The Database of Integer Sequences, Part 7 

     Part of the On-Line Encyclopedia of Integer Sequences


This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
    index.html:         Lookup 
    indexfr.html:       Francais
    demo1.html:         Demos 
    Sindx.html:         Index 
    WebCam.html:        WebCam
    Submit.html:        Contribute new sequence or comment 
    eishelp1.html:      Internal format 
    eishelp2.html:      Beautified format
    transforms.html:    Transforms 
    Spuzzle.html:       Puzzles 
    Shot.html:          Hot 
    classic.html:       Classics
    ol.html:            Superseeker 
    JIS/index.html:     Journal of Integer Sequences
    pages.html:         More pages 

Maintained  by:         N. J. A. Sloane (njas@research.att.com), 
    home page:          www.research.att.com/~njas/
    


(start)



%I A137596
%S A137596 1,1,1,1,2,1,1,3,4,1,1,4,11,7,1,1,5,26,32,11,1,1,6,57,122,76,16,1,1,7,
%T A137596 120,423,426,156,22,1,1,8,247,1389,2127,1206,288,29,1,1,9,502,4414,9897,
%U A137596 8157,2934,491,37,1
%N A137596 Triangle read by rows: A000012 * A048993.
%C A137596 Row sums = A005001 starting (1, 2, 4, 9, 24, 76, 279,...).
%F A137596 A000012 * A048993 as infinite lower triangular matrices.
%e A137596 First few rows of the triangle are:
%e A137596 1;
%e A137596 1, 1;
%e A137596 1, 2, 1;
%e A137596 1, 3, 4, 1;
%e A137596 1, 4, 11, 7, 1;
%e A137596 1, 5, 26, 32, 11, 1;
%e A137596 1, 6, 57, 122, 76, 16, 1;
%e A137596 1, 7, 120, 423, 426, 156, 22, 1;
%e A137596 ...
%Y A137596 Cf. A005001, A048993.
%Y A137596 Adjacent sequences: A137593 A137594 A137595 this_sequence A137597 A137598 A137599
%Y A137596 Sequence in context: A073165 A137153 A063841 this_sequence A111669 A124834 A104495
%K A137596 nonn,tabl
%O A137596 1,5
%A A137596 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 29 2008

%I A111669
%S A111669 1,1,1,1,2,1,1,3,4,1,1,4,11,7,1,1,5,26,32,12,1,1,6,57,122,92,20,1,1,7,
%T A111669 120,423,582,252,33,1,1,8,247,1389,3333,2598,681,54,1,1,9,502,4414,
%U A111669 18054,24117,11451,1815,88,1,1,10,1013,13744,94684,210990,172980,49566
%N A111669 Triangle read by rows, based on a simple Fibonacci recursion rule.
%C A111669 Subdiagonal is A000071(n+3). Row sums of inverse are 0^n.
%F A111669 Number triangle T(n, k)=T(n-1, k-1)+F(k+1)*T(n-1, k) where F(n)=A000045(n); Column k has g.f. x^k/Product(1-F(j+1)x, j, 0, k).
%e A111669 Triangle begins
%e A111669 1....1....2....3....5....8...13....F(k+1)
%e A111669 1
%e A111669 1....1
%e A111669 1....2....1
%e A111669 1....3....4....1
%e A111669 1....4...11....7....1
%e A111669 1....5...26...32...12....1
%e A111669 1....6...57..122...92...20....1
%e A111669 For example, T(6,3)=122=26+3*32=T(5,2)+F(4)*T(5,3)
%Y A111669 Cf. A111577, A111578, A111579, A008277, A039755.
%Y A111669 Adjacent sequences: A111666 A111667 A111668 this_sequence A111670 A111671 A111672
%Y A111669 Sequence in context: A137153 A063841 A137596 this_sequence A124834 A104495 A093541
%K A111669 easy,nonn,tabl
%O A111669 0,5
%A A111669 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 14 2005
%E A111669 Edited by Paul Barry, Nov 14 2005

%I A124834
%S A124834 1,1,1,1,2,1,1,3,4,1,1,4,11,8,1,1,5,26,42,16,1,1,6,57,184,163,32,1,1,7,
%T A124834 120,731,1358,638,64,1,1,8,247,2736,10121,10244,2510,128,1,1,9,502,9844,
%U A124834 70436,145475,78320,9908,256,1,1,10,1013,34448,468735,1911956,2141835
%N A124834 Triangle, read by rows, where the g.f. of column k, C_k(x), is equal to the product: C_k(x) = Product_{k=0..n} 1/(1 - binomial(n,k)*x).
%F A124834 T(n+1,n) = 2^n. T(n+2,n) = A032443(n) = Sum_{i=0..n} binomial(2*n,i).
%e A124834 Column g.f.s begin:
%e A124834 C_0(x) = 1/(1-x);
%e A124834 C_1(x) = 1/((1-x)(1-x));
%e A124834 C_2(x) = 1/((1-x)(1-2x)(1-x));
%e A124834 C_3(x) = 1/((1-x)(1-3x)(1-3x)(1-x));
%e A124834 C_4(x) = 1/((1-x)(1-4x)(1-6x)(1-4x)(1-x)); ...
%e A124834 Triangle begins:
%e A124834 1;
%e A124834 1, 1;
%e A124834 1, 2, 1;
%e A124834 1, 3, 4, 1;
%e A124834 1, 4, 11, 8, 1;
%e A124834 1, 5, 26, 42, 16, 1;
%e A124834 1, 6, 57, 184, 163, 32, 1;
%e A124834 1, 7, 120, 731, 1358, 638, 64, 1;
%e A124834 1, 8, 247, 2736, 10121, 10244, 2510, 128, 1;
%e A124834 1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1;
%e A124834 1, 10, 1013, 34448, 468735, 1911956, 2141835, 604160, 39203, 512, 1; ...
%o A124834 (PARI) {T(n,k)=polcoeff(1/prod(j=0,k,1-binomial(k,j)*x +x*O(x^n)),n-k)}
%Y A124834 Cf. A124835 (row sums), A124836 (central terms).
%Y A124834 Adjacent sequences: A124831 A124832 A124833 this_sequence A124835 A124836 A124837
%Y A124834 Sequence in context: A063841 A137596 A111669 this_sequence A104495 A093541 A089940
%K A124834 nonn,tabl
%O A124834 0,5
%A A124834 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 09 2006

%I A104495
%S A104495 1,1,1,1,2,1,1,3,4,1,1,4,12,5,1,1,5,34,17,7,1,1,6,98,51,32,8,1,1,7,294,
%T A104495 149,124,40,10,1,1,8,919,443,448,164,61,11,1,1,9,2974,1362,1576,612,298,72,
%U A104495 13,1,1,10,9891,4336,5510,2188,1294,370,99,14,1,1,11,33604,14227,19322,7698
%V A104495 1,-1,1,1,-2,1,-1,3,-4,1,1,-4,12,-5,1,-1,5,-34,17,-7,1,1,-6,98,-51,32,-8,1,-1,7,-294,
%W A104495 149,-124,40,-10,1,1,-8,919,-443,448,-164,61,-11,1,-1,9,-2974,1362,-1576,612,-298,72,
%X A104495 -13,1,1,-10,9891,-4336,5510,-2188,1294,-370,99,-14,1,-1,11,-33604,14227,-19322,7698
%N A104495 Matrix inverse of triangle A099602, read by rows, where row n of A099602 equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).
%C A104495 Row sums are A104496. Absolute row sums form A014137 (partial sums of Catalan numbers). Column 2 is signed A014143.
%F A104495 G.f.: A(x, y) = (1 + x*y/(1+x))/(1+x - x^2*y^2*Catalan(-x)^2), also G.f.: Column_k(x) = Catalan(-x)^(2*[k/2])/(1+x)^[(k+3)/2], where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).
%e A104495 Rows begin:
%e A104495 1;
%e A104495 -1,1;
%e A104495 1,-2,1;
%e A104495 -1,3,-4,1;
%e A104495 1,-4,12,-5,1;
%e A104495 -1,5,-34,17,-7,1;
%e A104495 1,-6,98,-51,32,-8,1;
%e A104495 -1,7,-294,149,-124,40,-10,1;
%e A104495 1,-8,919,-443,448,-164,61,-11,1;
%e A104495 -1,9,-2974,1362,-1576,612,-298,72,-13,1; ...
%o A104495 (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( (1+X*Y/(1+X))/(1+X-Y^2*(1-(1+4*X)^(1/2))^2/4),n,x),k,y)}
%Y A104495 Cf. A099602, A027907, A000108, A104496, A014137, A014143.
%Y A104495 Adjacent sequences: A104492 A104493 A104494 this_sequence A104496 A104497 A104498
%Y A104495 Sequence in context: A137596 A111669 A124834 this_sequence A093541 A089940 A123974
%K A104495 sign,tabl
%O A104495 0,5
%A A104495 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2005

%I A093541
%S A093541 1,1,1,1,2,1,1,3,4,1,1,4,12,6,1,1,5,28,33,8,1,1,6,56,138,72,10,1,1,7,
%T A093541 100,486,500,137,12,1,1,8,164,1498,2888,1532,236,14,1,1,9,252,4111,
%U A093541 14792,13772,4196,377,16,1,1,10,368,10210,67692,110856,57560,10518,568
%N A093541 Square array, read by antidiagonals, where column (k+1) equals the self-convolution of row k, with row 0 and column 0 consisting of all 1's.
%C A093541 Antidiagonal sums form A093542. Main diagonal is A093543.
%F A093541 T(n, k) = sum_{i=0..n} T(k-1, i)*T(k-1, n-i), with T(n, 0)=T(0, k)=1 for n>=0, k>0.
%e A093541 Column 2: {1,4,12,28,56,100,...} equals the self-convolution of row 1: {1,2,4,6,8,10,...}.
%e A093541 Rows begin:
%e A093541 [1,1,1,1,1,1,1,1,1,1,1,1,1,...],
%e A093541 [1,2,4,6,8,10,12,14,16,18,20,22,...],
%e A093541 [1,3,12,33,72,137,236,377,568,817,...],
%e A093541 [1,4,28,138,500,1532,4196,10518,...],
%e A093541 [1,5,56,486,2888,13772,57560,219834,...],
%e A093541 [1,6,100,1498,14792,110856,698816,...],
%e A093541 [1,7,164,4111,67692,812492,7930308,...],
%e A093541 [1,8,252,10210,278396,5364868,...],
%e A093541 [1,9,368,23288,1040856,31939300,...],
%e A093541 [1,10,516,49394,3581120,173226000,...],...
%o A093541 (PARI) T(n,k)=if(n<0|k<0,0,if(n==0|k==0,1,sum(i=0,n,T(k-1,i)*T(k-1,n-i))))
%Y A093541 Cf. A093542, A093543.
%Y A093541 Adjacent sequences: A093538 A093539 A093540 this_sequence A093542 A093543 A093544
%Y A093541 Sequence in context: A111669 A124834 A104495 this_sequence A089940 A123974 A056863
%K A093541 nonn,tabl
%O A093541 0,5
%A A093541 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2004

%I A089940
%S A089940 1,1,1,2,1,1,3,4,1,1,6,5,6,1,1,10,15,7,8,1,1,20,21,28,9,10,1,1,35,56,36,
%T A089940 45,11,12,1,1,70,84,120,55,66,13,14,1,1,126,210,165,220,78,91,15,16,1,1,
%U A089940 252,330,495,286,364,105,120,17,18,1,1,462,792,715,1001,455,560,136,153
%N A089940 Triangle read by rows: T(n,k)=binomial(n+k,floor((n-k)/2))
%Y A089940 Columns include A001405, A037955, A037956, A037957. Row sums are A089941.
%Y A089940 Adjacent sequences: A089937 A089938 A089939 this_sequence A089941 A089942 A089943
%Y A089940 Sequence in context: A124834 A104495 A093541 this_sequence A123974 A056863 A120019
%K A089940 easy,nonn,tabl
%O A089940 0,4
%A A089940 Paul Barry (pbarry(AT)wit.ie), Nov 16 2003

%I A123974
%S A123974 1,1,1,0,2,1,1,3,4,1,3,6,14,7,1,14,24,72,48,12,1,109,172,586,449,143,
%T A123974 20,1,1403,2103,7718,6375,2296,402,33,1,29354,42588,163595,141144,54448,
%U A123974 10718,1094,54,1,996633,1416535,5597100,4956116,1990080,418458,47881,2929,88,1
%V A123974 1,1,-1,0,-2,1,-1,-3,4,-1,-3,-6,14,-7,1,-14,-24,72,-48,12,-1,-109,-172,586,-449,143,
%W A123974 -20,1,-1403,-2103,7718,-6375,2296,-402,33,-1,-29354,-42588,163595,-141144,54448,
%X A123974 -10718,1094,-54,1,-996633,-1416535,5597100,-4956116,1990080,-418458,47881,-2929,88,-1
%N A123974 Fibonacci central tridiagonal matrices as a triangular sequence from a recursive polynomial definition.
%C A123974 Matrices: {{1}}, {{1, -1}, {-1, 1}}, {{1, -1, 0}, {-1, 1, -1}, {0, -1, 2}}, {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 2, -1}, {0, 0, -1, 3}}, {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 2, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 5}}, {{1, -1, 0, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {0, -1, 2, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, {0, 0, 0, -1, 5, -1}, {0, 0, 0, 0, -1, 8}} The Dombrowski paper defines a recursive polynomial form from the tridiagonal matrices: p[1,x]=1,p[2,x]=(x-b[1])/a[1] p[n,x]=((x-b[n-1])*p[n-1,x]-a[n-2]*p[n-2,x])/a[n-1] As long as b[n-1]/a[n-1] and a[n-2]/a[n-1] behave well ( rationally or like Integers) this definition is a good recursive polynomial on a tridiagonal matrix. Here I use: a[n]=-1 and b[n]=Fibonacci[n]
%D A123974 Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334
%F A123974 M(n,m)=If[ n == m, Fibonacci[n], If[n == m - 1 || n == m + 1, -1, 0]]
%e A123974 Triangular sequence:
%e A123974 {1},
%e A123974 {1, -1},
%e A123974 {0, -2, 1},
%e A123974 {-1, -3, 4, -1},
%e A123974 {-3, -6, 14, -7, 1},
%e A123974 {-14, -24, 72, -48, 12, -1},
%e A123974 {-109, -172, 586, -449, 143, -20, 1},
%e A123974 {-1403, -2103, 7718, -6375,2296, -402, 33, -1},
%e A123974 {-29354, -42588, 163595, -141144, 54448, -10718, 1094, -54, 1}
%t A123974 T[n_, m_] := If[ n == m, Fibonacci[n], If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]; Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]
%Y A123974 Adjacent sequences: A123971 A123972 A123973 this_sequence A123975 A123976 A123977
%Y A123974 Sequence in context: A104495 A093541 A089940 this_sequence A056863 A120019 A128314
%K A123974 uned,probation,sign
%O A123974 1,5
%A A123974 Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 30 2006

%I A056863
%S A056863 1,1,1,2,1,1,3,4,2,1,4,9,10,4,1,5,16,28,24,8,1,6,25,60,80,56,16
%V A056863 1,-1,1,-2,1,1,-3,4,2,1,-4,9,10,4,1,-5,16,28,24,8,1,-6,25,60,80,56,16
%N A056863 Related to triangle of number of rises in set partitions of n at a given index i.
%C A056863 Number of rises in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1, and s(i) <= 1 + max of previous s(j)'s.
%D A056863 W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.
%e A056863 For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3, and i = 5.
%e A056863 1,-1; 1,-2,1; 1,-3,4,2; 1,-4,9,10,4; ...
%Y A056863 Cf. Bell numbers A000110.
%Y A056863 Cf. A056857-A056862.
%Y A056863 Adjacent sequences: A056860 A056861 A056862 this_sequence A056864 A056865 A056866
%Y A056863 Sequence in context: A093541 A089940 A123974 this_sequence A120019 A128314 A025564
%K A056863 easy,sign,tabl,more
%O A056863 1,4
%A A056863 Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

%I A120019
%S A120019 1,1,1,1,2,1,1,3,4,2,1,4,9,10,6,1,5,16,30,32,18,1,6,25,68,114,116,53,1,
%T A120019 7,36,130,312,480,440,158,1,8,49,222,710,1536,2157,1708,481,1,9,64,350,
%U A120019 1416,4070,8000,10092,6760,1491,1,10,81,520,2562,9348,24365,43472,48525
%N A120019 Square table, read by antidiagonals, of self-compositions of A120010.
%C A120019 The g.f. of row n is the composition: (1-sqrt(1-4*x))/2 o x/(1-nx) o (x-x^2).
%F A120019 T(n, k) = Sum_{j=1..k}Catalan(k-j)*[Sum_{i=1..j}(-1)^(j-i)*n^(i-1)*C(k-j+i, j-i)*C(k-j+i-1, i-1)]; Also, T(n, k) = Sum_{j=0..k-1}n^j*[Sum_{i=j..k-1}(-1)^(i-j)*Catalan(k-i-1)*C(k-i+j, i-j)*C(k-i+j-1, j)]; where Catalan(n) = A000108(n) = C(2n, n)/(n+1).
%e A120019 Square table begins:
%e A120019 1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, ...
%e A120019 1, 2, 4, 10, 32, 116, 440, 1708, 6760, 27232, ...
%e A120019 1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, ...
%e A120019 1, 4, 16, 68, 312, 1536, 8000, 43472, 243808, 1400448, ...
%e A120019 1, 5, 25, 130, 710, 4070, 24365, 151330, 968785, 6355795, ...
%e A120019 1, 6, 36, 222, 1416, 9348, 63768, 448188, 3234216, 23875296, ...
%e A120019 1, 7, 49, 350, 2562, 19236, 148085, 1167488, 9409645, 77367087, ...
%e A120019 1, 8, 64, 520, 4304, 36320, 312512, 2740672, 24476800, 222358528, ...
%e A120019 1, 9, 81, 738, 6822, 64026, 610245, 5906502, 58033953, 578488563, ...
%e A120019 1, 10, 100, 1010, 10320, 106740, 1117880, 11855660, 127313320, ...
%e A120019 Successive self-compositions of F(x), the g.f. of A120010, begin:
%e A120019 F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
%e A120019 F(F(x)) = x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
%e A120019 F(F(F(x))) = x + 3x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
%e A120019 F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 68x^4 + 312x^5 + 1536x^6 +...
%e A120019 F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + 710x^5 + 4070x^6 +...
%e A120019 F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + 9348x^6 +...
%o A120019 (PARI) {T(n,k)=sum(j=1, k, binomial(2*k-2*j, k-j)/(k-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(k-j+i, j-i)*binomial(k-j+i-1, i-1)*n^(i-1)))}
%Y A120019 Rows: A120010, A120017, A120018; Diagonals: A120020, A120021. Variant: A120013.
%Y A120019 Adjacent sequences: A120016 A120017 A120018 this_sequence A120020 A120021 A120022
%Y A120019 Sequence in context: A089940 A123974 A056863 this_sequence A128314 A025564 A052265
%K A120019 nonn,tabl
%O A120019 1,5
%A A120019 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 14 2006

%I A128314
%S A128314 1,1,1,1,1,1,1,1,2,1,1,3,4,3,1,1,3,7,7,4,1,1,3,8,13,11,5,1,1,1,10,21,24,
%T A128314 16,6,1,1,5,17,35,46,40,22,7,1,1,7,25,53,81,86,62,29,8,1
%V A128314 1,-1,1,-1,-1,1,-1,1,-2,1,-1,-3,4,-3,1,-1,3,-7,7,-4,1,-1,-3,8,-13,11,-5,1,-1,1,-10,21,
%W A128314 -24,16,-6,1,-1,-5,17,-35,46,-40,22,-7,1,-1,7,-25,53,-81,86,-62,29,-8,1
%N A128314 Triangle, A000012 * A128313.
%C A128314 Row sums = the Mertens sequence: A002321, (1, 0, -1, -1, -2,...).
%F A128314 A000012 * A128313 as infinite lower triangular matrices.
%e A128314 First few rows of the triangle are:
%e A128314 1;
%e A128314 -1, 1;
%e A128314 -1, -1, 1;
%e A128314 -1, 1, -2, 1
%e A128314 -1, -3, 4, -3, 1;
%e A128314 -1, 3, -7, 7, -4, 1;
%e A128314 -1, -3, 8, -13, 11, -5, 1;
%e A128314 ...
%Y A128314 Cf. A128313, A000012, A002321.
%Y A128314 Adjacent sequences: A128311 A128312 A128313 this_sequence A128315 A128316 A128317
%Y A128314 Sequence in context: A123974 A056863 A120019 this_sequence A025564 A052265 A055068
%K A128314 tabl,sign
%O A128314 1,9
%A A128314 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 25 2007

%I A025564
%S A025564 1,1,1,1,2,1,1,3,4,3,1,1,4,8,10,8,4,1,1,5,13,22,26,22,13,5,1,1,6,19,40,
%T A025564 61,70,61,40,19,6,1,1,7,26,65,120,171,192,171,120,65,26,7,1,1,8,34,98,
%U A025564 211,356,483,534,483,356,211,98,34,8,1,1,9,43,140,343,665,1050,1373
%N A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.
%C A025564 T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k)=n and the string does not the substring "00". E.g. T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112 - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007
%F A025564 T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 2, 1], [1, 3, 4, 3, 1].
%F A025564 G.f.: (1+yz)/[1-z(1+y+y^2)].
%e A025564 ..........1
%e A025564 .......1..1..1
%e A025564 ....1..3..4..3..1
%e A025564 ..1.4..8..10.8..4..1
%e A025564 1.5.13.22.26.22.13.5.1
%o A025564 (PARI) T(n,k)=if(n<0||k<0||k>2*n,0,if(n==0,1,if(n==1,[1,2,1][k+1],if(n==2,[1,3,4,3,1][k+1],T(n-1,k-2)+T(n-1,k-1)+T(n-1,k)))))
%o A025564 (PARI) T(n,k)=polcoeff(Ser(polcoeff(Ser((1+y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)
%o A025564 (PARI) {T(n,k)= if(n<0||k<0||k>2*n, 0, if(n==0, 1, polcoeff( (1+x+x^2)^n, k)+ polcoeff( (1+x+x^2)^(n-1), k-1)))}
%Y A025564 Columns include A025565, A025566, A025567, A025568.
%Y A025564 Cf. A025177.
%Y A025564 Adjacent sequences: A025561 A025562 A025563 this_sequence A025565 A025566 A025567
%Y A025564 Sequence in context: A056863 A120019 A128314 this_sequence A052265 A055068 A015138
%K A025564 nonn,tabf,easy
%O A025564 1,5
%A A025564 Clark Kimberling (ck6(AT)evansville.edu)
%E A025564 Edited by Ralf Stephan, Jan 09 2005

%I A052265
%S A052265 1,1,1,2,1,1,3,4,3,1,1,4,9,16,20,16,9,4,1,1,5,17,52,136,284,477,655,
%T A052265 730,655,477,284,136,52,17,5,1,1,6,28,134,625,2674,10195,34230,100577,
%U A052265 258092,579208,1140090,1974438,3016994,4077077,4881092,5182326,4881092
%N A052265 Triangle giving a(n,r) = number of equivalence classes of Boolean functions of n variables and range r=0..2^n under action of symmetric group.
%D A052265 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 147.
%H A052265 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean">Index entries for sequences related to Boolean functions</a>
%e A052265 [1, 1], [1, 2, 1], [1, 3, 4, 3, 1], [1, 4, 9, 16, 20, 16, 9, 4, 1], ...
%Y A052265 Row sums give A003180.
%Y A052265 Adjacent sequences: A052262 A052263 A052264 this_sequence A052266 A052267 A052268
%Y A052265 Sequence in context: A120019 A128314 A025564 this_sequence A055068 A015138 A100529
%K A052265 nonn,tabf,nice
%O A052265 0,4
%A A052265 Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 04 2000

%I A055068
%S A055068 1,1,1,1,2,1,1,3,4,3,1,4,10,24,10,1,5,20,105,160,105,1,6,35,336,1260,
%T A055068 3360,1260,1,7,56,882,6720,48510,80640,48510,1,8,84,2016,27720,443520,
%U A055068 2162160,6209280,2162160,1,9,120,4158,95040,2972970,34594560,312161850
%N A055068 Triangular array for David G. Cantor's sigma function.
%D A055068 David G. Cantor, On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. 447 (1994), 91-145.
%F A055068 T(n, k)*T(n-2, k-1)-2*T(n-1, k-1)*T(n-1, k)+T(n, k-1)*T(n-2, k)=0.
%e A055068 rows: 1; 1,1; 1,2,1; 1,3,4,3; 1,4,10,24,10; 1,5,20,105,160,105; ...
%o A055068 (PARI) {T(n, k)= if(k<0|k>n, 0, prod(i=1, (k+1)\2, binomial(n+2*i-1-k%2, 4*i-1-k%2*2)))}
%Y A055068 Adjacent sequences: A055065 A055066 A055067 this_sequence A055069 A055070 A055071
%Y A055068 Sequence in context: A128314 A025564 A052265 this_sequence A015138 A100529 A124424
%K A055068 nonn,tabl,easy
%O A055068 0,5
%A A055068 Michael Somos

%I A015138
%S A015138 1,1,1,2,1,1,3,4,3,1,11,5,6,3,4,8,8,13,19,5,27,11,11,21,5,6,9,15,58,4,
%T A015138 31,16,24,32,12,53,18,76,24,21,42,27,21,59,12,44,69,93,21,5,32,30,26,36,
%U A015138 145,63,40,58,59,20,62,124,72,32,24,24,33,128,99,12,71,213,36,18,20,316
%N A015138 Consider Fibonacci-type sequences b(0)=X, b(1)=Y, b(k)=b(k-1)+b(k-2) mod n; all are periodic; sequence gives number of maximal length periods.
%Y A015138 Adjacent sequences: A015135 A015136 A015137 this_sequence A015139 A015140 A015141
%Y A015138 Sequence in context: A025564 A052265 A055068 this_sequence A100529 A124424 A057044
%K A015138 nonn
%O A015138 1,4
%A A015138 Phil Carmody (pc+oeis(AT)asdf.org)
%E A015138 More terms from Larry Reeves (larryr(AT)acm.org), Jan 06 2005

%I A100529
%S A100529 1,1,1,1,2,1,1,3,4,3,4,2,2,1,1,12,15,13,14,11,12,9,10,6,6,4,4,2,2,1,
%T A100529 1,84,91,82,89,77,80,70,73,60,63,53,54,43,44,35,36,26,26,20,20,14,
%U A100529 14,10,10,6,6,4,4,2,2,1,1,908
%N A100529 a(n) = minimal k such that n has a partition into k parts with the property that every number <= m can be partitioned into a subset of these parts.
%D A100529 E. O'Shea, M-partitions: optimal partitions of weight for one scale pan, Discrete Math 289 (2004), 81-93.
%D A100529 O. J. Rodseth, Enumeration of M-partitions, Discrete Math., 306 (2006), 694-698.
%F A100529 If 2^m + 2^(m-1) - 1 <= n <= 2^(m+1) - 1 for some m, let i = 2^(m+1) - 1 - n. Then a(n) = A000123([i/2]). This determines half the values.
%Y A100529 Cf. A000123 (binary partitions), A002033 (perfect partitions).
%Y A100529 Adjacent sequences: A100526 A100527 A100528 this_sequence A100530 A100531 A100532
%Y A100529 Sequence in context: A052265 A055068 A015138 this_sequence A124424 A057044 A068098
%K A100529 nonn
%O A100529 1,5
%A A100529 njas, Dec 31 2004

%I A124424
%S A124424 1,0,1,1,0,1,1,2,1,1,3,4,5,2,1,7,14,16,10,4,1,25,48,61,42,20,6,1,79,194,
%T A124424 250,200,106,38,9,1,339,820,1145,958,569,230,66,12,1,1351,3794,5554,
%U A124424 5096,3251,1486,486,112,16,1,6721,18960,29101,28010,19110,9470,3477,930
%N A124424 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting of entries of the same parity (0<=k<=n).
%C A124424 Row sums are the Bell numbers (A000110). T(n,0)=A124425(n).
%F A124424 The generating polynomial of row n is P[n](t)=Q[n](t,t,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
%e A124424 T(4,2)=5 because we have 13|24, 14|2|3, 1|2|34, 1|23|4, and 12|3|4.
%e A124424 Triangle starts:
%e A124424 1;
%e A124424 0,1;
%e A124424 1,0,1;
%e A124424 1,2,1,1;
%e A124424 3,4,5,2,1;
%e A124424 7,14,16,10,4,1;
%p A124424 Q[0]:=1: for n from 1 to 11 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 11 do P[n]:=sort(subs({s=t,x=1},Q[n])) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A124424 Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124425.
%Y A124424 Adjacent sequences: A124421 A124422 A124423 this_sequence A124425 A124426 A124427
%Y A124424 Sequence in context: A055068 A015138 A100529 this_sequence A057044 A068098 A135722
%K A124424 nonn,tabl
%O A124424 0,8
%A A124424 Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 01 2006

%I A057044
%S A057044 2,1,1,3,4,5,4,8,9,3,14,12,4,8,19,15,5,85,1,105,99,56,19,183,59,150,
%T A057044 511,250,382,36,988,1937,1035,1240,1733,3862,1425,4295,229,8648,14795,
%U A057044 11628
%N A057044 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; each k is an R(i(k),j(k)) and A057044(n)=j(L(n)), where L(n) is the n-th Lucas number.
%Y A057044 Adjacent sequences: A057041 A057042 A057043 this_sequence A057045 A057046 A057047
%Y A057044 Sequence in context: A015138 A100529 A124424 this_sequence A068098 A135722 A049513
%K A057044 nonn
%O A057044 1,1
%A A057044 Clark Kimberling (ck6(AT)evansville.edu), Jul 30 2000

%I A068098
%S A068098 2,1,1,3,4,11,47,521,24476,12752043,312119004989,3980154972736918051,
%T A068098 1242282009792667284144565908482
%N A068098 a(n) = L(F(n)) where F(n) = n-th Fibonacci number and L(n) = n-th Lucas number.
%F A068098 For n>2, a(n) = round{Phi^F(n-1)}. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 19 2004
%Y A068098 Adjacent sequences: A068095 A068096 A068097 this_sequence A068099 A068100 A068101
%Y A068098 Sequence in context: A100529 A124424 A057044 this_sequence A135722 A049513 A121207
%K A068098 easy,nonn
%O A068098 0,1
%A A068098 Leroy Quet (qq-quet(AT)mindspring.com), Mar 22 2002
%E A068098 More terms from Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 19 2004

%I A135722
%S A135722 1,1,1,1,1,2,1,1,3,5,1,1,3,15,14,1,1,3,23,84,42,1,1,3,27,244,466,132,1,
%T A135722 1,3,28,494,2444,2514,429,1,1,3,28,796,8720,22022,13233,1430,1,1,3,28,
%U A135722 1094,24394,128514,181841,68376,4862
%N A135722 A000012 * A122890.
%C A135722 Row sums = A003422: (1, 2, 4, 10, 34, 154, 874,...). Main diagonal = A000108, the Catalan numbers: (1, 1, 2, 5, 14, 42, 132,...)
%F A135722 A000012 * A122890 as infinite lower triangular matrices.
%e A135722 First few rows of the triangle are:
%e A135722 1;
%e A135722 1, 1;
%e A135722 1, 1, 2;
%e A135722 1, 1, 3, 5;
%e A135722 1, 1, 3, 15, 14;
%e A135722 1, 1, 3, 23, 84, 42;
%e A135722 1, 1, 3, 27, 244, 466, 132;
%e A135722 1, 1, 3, 28, 494, 2444, 2514, 429;
%e A135722 ...
%Y A135722 Cf. A003422, A122890, A000108.
%Y A135722 Adjacent sequences: A135719 A135720 A135721 this_sequence A135723 A135724 A135725
%Y A135722 Sequence in context: A124424 A057044 A068098 this_sequence A049513 A121207 A097084
%K A135722 nonn,tabl
%O A135722 0,6
%A A135722 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 25 2007

%I A049513
%S A049513 1,1,1,1,2,1,1,3,5,1,1,4,9,13,1,1,5,13,25,33,1,1,6,17,37,65,81,1,1,7,
%T A049513 21,49,97,161,193,1,1,8,25,61,129,241,385,449,1,1,9,29,73,161,321,577,
%U A049513 897,1025,1,1,10,33,85,193,401,769,1345,2049,2305,1,1,11,37,97,225,481
%N A049513 Array T by antidiagonals: T(k,n)=k*n*2^(n-1)+1, n >= 0, k >= 0.
%e A049513 Antidiagonals: 1; 1,1; 1,2,1; 1,3,5,1; 1,4,9,13,1; ...
%o A049513 (PARI) T(k,n)=k*n*2^(n-1)+1
%Y A049513 A005183(n)=T(1, n), A002064(n)=T(2, n), A048474(n)=T(3, n), A000337(n)=T(4, n), A016813(n)=T(n, 2), A017533(n)=T(n, 3). Cf. A049069, A048472.
%Y A049513 Essentially the same as A049069.
%Y A049513 Adjacent sequences: A049510 A049511 A049512 this_sequence A049514 A049515 A049516
%Y A049513 Sequence in context: A057044 A068098 A135722 this_sequence A121207 A097084 A094954
%K A049513 nonn,tabl,easy
%O A049513 0,5
%A A049513 Michael Somos

%I A121207
%S A121207 1,1,1,1,1,2,1,1,3,5,1,1,4,9,15,1,1,5,14,31,52,1,1,6,20,54,121,203,
%T A121207 1,1,7,27,85,233,523,877,1,1,8,35,125,400,1101,2469,4140,1,1,9,44,
%U A121207 175,635,2046,5625,12611,21147,1,1,10,54,236,952,3488,11226,30846
%N A121207 Triangle read by rows. The definition is by diagonals. The r-th diagonal from the right, for r >= 0, is given by b(0) = b(1) = 1; b(n+1) = Sum_{k=0..n} binomial(n+2,k+r)*a(k).
%C A121207 Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Dec 12 2006. (Start) Consider the row reversal, which is A124496 with an additional left column (A000110 = Bell numbers). The matrix inverse of this triangle is very simple:
%C A121207 1;
%C A121207 -1, 1;
%C A121207 -1, -1, 1;
%C A121207 -1, -2, -1, 1;
%C A121207 -1, -3, -3, -1, 1;
%C A121207 -1, -4, -6, -4, -1, 1;
%C A121207 -1, -5, -10, -10, -5, -1, 1;
%C A121207 -1, -6, -15, -20, -15, -6, -1, 1;
%C A121207 -1, -7, -21, -35, -35, -21, -7, -1, 1;
%C A121207 -1, -8, -28, -56, -70, -56, -28, -8, -1, 1; ...
%C A121207 This gives the recurrence and explains why the Bell numbers appear. (End)
%e A121207 Triangle begins:
%e A121207 1,
%e A121207 1, 1,
%e A121207 1, 1, 2,
%e A121207 1, 1, 3, 5,
%e A121207 1, 1, 4, 9, 15,
%e A121207 1, 1, 5, 14, 31, 52,
%e A121207 1, 1, 6, 20, 54, 121, 203,
%e A121207 1, 1, 7, 27, 85, 233, 523, 877,
%e A121207 1, 1, 8, 35, 125, 400,1101,2469,4140,
%e A121207 1, 1, 9, 44, 175, 635,2046,5625,12611,21147,
%e A121207 1, 1, 10, 54, 236, 952,3488,11226,30846,69161,115975,
%e A121207 1, 1, 11, 65, 309,1366,5579,20425,65676,180474,404663,678570,
%e A121207 1, 1, 12, 77, 395,1893,8494,34685,126817,407787,1120666,2512769,4213597,
%e A121207 1, 1, 13, 90, 495,2550,12432,55818,227550,831915,2675410,7352471,16485691,27644437, etc
%p A121207 (Maple program from R. J. Mathar) Gould := proc(n,d) local k; if n<=1 then RETURN(1); else
%p A121207 # This is the Jovovic formula with general index 'd'
%p A121207 # where A040027, A045499 etc. use one explicit integer
%p A121207 # Index n+1 is shifted to n from the original formula.
%p A121207 RETURN(add(binomial(n-1+d,k+d)*Gould(k,d),k=0..n-1));
%p A121207 fi;
%p A121207 end:
%p A121207 # row and col refer to the extrapolated super-table:
%p A121207 for row from 0 to 13 do
%p A121207 # working up to row, not row-1, shows also the Bell numbers
%p A121207 # at the end of each row
%p A121207 for col from 0 to row do
%p A121207 # 'diag' is constant for one of A040027, A045499 etc
%p A121207 diag := row-col;
%p A121207 printf("%4d,",Gould(col,diag));
%p A121207 od;
%p A121207 print();
%p A121207 od;
%Y A121207 Diagonals, reading from the right, are A000110, A040027, A045501, A045499, A045500.
%Y A121207 A124496 is a very similar triangle, obtained by reversing the rows and appending a right-most diagonal which is A000110, the Bell numbers.
%Y A121207 Adjacent sequences: A121204 A121205 A121206 this_sequence A121208 A121209 A121210
%Y A121207 Sequence in context: A068098 A135722 A049513 this_sequence A097084 A094954 A083064
%K A121207 nonn,tabl
%O A121207 0,6
%A A121207 njas, based on email from R. J. Mathar, Dec 11 2006

%I A097084
%S A097084 1,1,1,1,2,1,1,3,5,1,1,4,10,10,1,1,5,18,28,17,1,1,6,27,74,69,26,1,1,7,
%T A097084 39,137,245,151,37,1,1,8,52,236,586,676,298,50,1,1,9,68,372,1194,2126,
%U A097084 1634,540,65,1,1,10,85,552,2322,5152,6620,3578,913,82,1,1,11,105,777
%N A097084 Triangle, read by rows, where the n-th diagonal equals the n-th row transformed by triangle A008459 (squared binomial coefficients).
%C A097084 Row sums form A097085.
%F A097084 T(n, k) = Sum_{j=0..k} T(n-k, j)*C(k, j)^2.
%e A097084 T(8,3) = 236 = (1)*1^2 + (5)*3^2 + (18)*3^2 + (28)*1^2
%e A097084 = Sum_{j=0..3} T(5,j)*C(3,j)^2.
%e A097084 Rows begin:
%e A097084 [1],
%e A097084 [1,1],
%e A097084 [1,2,1],
%e A097084 [1,3,5,1],
%e A097084 [1,4,10,10,1],
%e A097084 [1,5,18,28,17,1],
%e A097084 [1,6,27,74,69,26,1],
%e A097084 [1,7,39,137,245,151,37,1],
%e A097084 [1,8,52,236,586,676,298,50,1],...
%o A097084 (PARI) T(n,k)=if(n<k|k<0,0,if(n==k|k==0,1,sum(j=0,n-k,T(n-k,j)*binomial(k,j)^2)))
%Y A097084 Cf. A097085, A008459.
%Y A097084 Adjacent sequences: A097081 A097082 A097083 this_sequence A097085 A097086 A097087
%Y A097084 Sequence in context: A135722 A049513 A121207 this_sequence A094954 A083064 A112338
%K A097084 nonn,tabl
%O A097084 0,5
%A A097084 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 23 2004

%I A094954
%S A094954 1,1,1,1,2,1,1,3,5,1,1,4,11,13,1,1,5,19,41,34,1,1,6,29,91,153,89,1,1,7,
%T A094954 41,169,436,571,233,1,1,8,55,281,985,2089,2131,610,1,1,9,71,433,1926,
%U A094954 5741,10009,7953,1597,1,1,10,89,631,3409,13201,33461,47956,29681
%N A094954 Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1.
%C A094954 Also, values of polynomials with coefficients in A098493 (see Fink et al.). See A098495 for negative k.
%C A094954 Number of dimer tilings of the graph S_{k-1} X P_{2n-2}.
%D A094954 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%H A094954 Elizabeth Wilmer, <a href="http://www.oberlin.edu/math/faculty/wilmer/OEISconj87.pdf">A note on Stephan's conjecture 87</a>
%F A094954 Recurrence: T(k, 1) = 1, T(k, 2) = k-1, T(k, n) = kT(k, n-1) - T(k, n-2).
%F A094954 For n>3, T(k, n) = [k(k-2) + T(k, n-1)T(k, n-2)] / T(k, n-3).
%F A094954 T(k, n+1) = S(n, k) - S(n-1, k) = U(n, k/2) - U(n-1, k/2), with S, U = Chebyshev polynomials of second kind.
%F A094954 T(k+2, n+1) = Sum[i=0..n, k^(n-i) * C(2n-i, i)] (from comments by Benoit Cloitre).
%e A094954 1,1,1,1,1,1,1,1,1,1,1,1,1,1, ...
%e A094954 1,2,5,13,34,89,233,610,1597, ...
%e A094954 1,3,11,41,153,571,2131,7953, ...
%e A094954 1,4,19,91,436,2089,10009,47956, ...
%e A094954 1,5,29,169,985,5741,33461,195025, ...
%e A094954 1,6,41,281,1926,13201,90481,620166, ...
%o A094954 (PARI) T(k,n)=polcoeff(x*(1-x)/(1-k*x+x*x),n)
%Y A094954 Rows are first differences of rows in array A073134.
%Y A094954 Rows 2-14 are A000012, A001519, A079935/A001835, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570. Other rows: A007805 (k=18), A075839 (k=20), A077420 (k=34), A078988 (k=66).
%Y A094954 Columns include A028387. Diagonals include A094955, A094956. Antidiagonal sums are A094957.
%Y A094954 Adjacent sequences: A094951 A094952 A094953 this_sequence A094955 A094956 A094957
%Y A094954 Sequence in context: A049513 A121207 A097084 this_sequence A083064 A112338 A111672
%K A094954 nonn,tabl
%O A094954 1,5
%A A094954 Ralf Stephan (ralf(AT)ark.in-berlin.de), May 31 2004

%I A083064
%S A083064 1,1,1,1,2,1,1,3,5,1,1,4,11,14,1,1,5,19,43,41,1,1,6,29,94,171,122,1,1,7,
%T A083064 41,173,469,683,365,1,1,8,55,286,1037,2344,2731,1094,1,1,9,71,439,2001,
%U A083064 6221,11719,10923,3281,1,1,10,89,638,3511,14006,37325,58594,43691,9842
%N A083064 Square number array T(n,k)=(k(k+2)^n+1)/(k+1) read by antidiagonals.
%e A083064 Rows begin
%e A083064 1 1 1 1 1 ...
%e A083064 1 2 5 14 41 ...
%e A083064 1 3 11 43 171 ...
%e A083064 1 4 19 94 469 ...
%e A083064 1 5 29 173 1037 ...
%Y A083064 Rows include A007583, A083065, A083066, A083067, A083068. Diagonals include A083069, A083070, A083071, A083072, A083073. Columns include A000027, A028387, A083074.
%Y A083064 Adjacent sequences: A083061 A083062 A083063 this_sequence A083065 A083066 A083067
%Y A083064 Sequence in context: A121207 A097084 A094954 this_sequence A112338 A111672 A128198
%K A083064 easy,nonn,tabl
%O A083064 0,5
%A A083064 Paul Barry (pbarry(AT)wit.ie), Apr 21 2003

%I A112338
%S A112338 1,1,1,1,2,1,1,3,5,1,1,4,12,14,1,1,5,22,57,42,1,1,6,35,148,303,132,1,1,
%T A112338 7,51,305,1144,1743,429,1,8,70,546,3105,9784,10629,1430,1
%N A112338 Triangle read by rows, generated from A001263.
%C A112338 Rows of the array are row sums of n-th powers of the Narayana triangle; e.g. row 1 = A000108: (1, 2, 5, 14, 42...); row 2 = row sums of the Narayana triangle squared (A103370): (1, 3, 12, 57, 303...), etc.
%C A112338 First few rows of the array are:
%C A112338 1, 1, 1, 1, 1, 1,...
%C A112338 1, 2, 5, 14, 42, 132,...
%C A112338 1, 3, 12, 57, 303, 1743,...
%C A112338 1, 4, 22, 148, 1144, 9784,...
%C A112338 1, 5, 35, 305, 3105, 35505,...
%C A112338 First few rows of the triangle are:
%C A112338 1;
%C A112338 1, 1;
%C A112338 1, 2, 1;
%C A112338 1, 3, 5, 1;
%C A112338 1, 4, 12, 14, 1;
%C A112338 1, 5, 22, 57, 42, 1;
%C A112338 1, 6, 35, 148, 303, 132, 1;
%F A112338 Let M = the infinite lower triangular Narayana triangle (A001263). Perform M^n * [1 0 0 0...] getting an array. Take antidiagonals of the array which become rows of the triangle A112338.
%e A112338 In the array, antidiagonal terms (1, 3, 5, 1) become row 3 of the triangle.
%Y A112338 Cf. A001263, A000326, A005915, A095266, A000108, A103370.
%Y A112338 Adjacent sequences: A112335 A112336 A112337 this_sequence A112339 A112340 A112341
%Y A112338 Sequence in context: A097084 A094954 A083064 this_sequence A111672 A128198 A123349
%K A112338 nonn,tabl
%O A112338 0,5
%A A112338 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 04 2005

%I A111672
%S A111672 1,1,1,1,2,1,1,3,5,1,1,4,12,15,1,1,5,22,60,52,1,1,6,35,154,358,203,1,1,
%T A111672 7,51,315,1304,2471,877,1,1,8,70,561,3455,12915,19302,21147,1
%N A111672 Triangle, antidiagonals are number of N-level labeled rooted trees with n leaves.
%C A111672 N-th row of the array generated from the Stirling number of the second kind triangle = the entry associated with "Number of N-level labeled rooted trees with n leaves". First few rows of the array are: 1, 1, 1, 1, 1, 1,... 1, 2, 5, 15, 52, 203,... 1, 3, 12, 60, 358, 2471,... 1, 4, 22, 154, 1304, 12915,... 1, 5, 35, 315, 3455, 44590,... 1, 6, 51, 561, 7556, 120196,... 1, 7, 70, 910, 14532, 274778,... Where the rows (prefaced with another "1" are): row 2, A000110; row 3, A000258; row 4, A000307, row 5, A000357; row 6, A000405; row 7, A001669, row 8, A081624...; and so on. By columns of the triangle A111672, column 3 (1, 5, 12, 22, 35...) = A000326, column 4 = A005945.
%F A111672 Let the Stirling number of the second kind triangle A008277 be an infinite lower triangular matrix M. Perform M^n * [1, 0, 0, 0...] getting an array. The antidiagonals of the array become the rows of A111672; while rows of the array become diagonals of A111672.
%e A111672 Terms 1, 3, 5, 1 of an antidiagonal in the array becomes row 4 of the triangle.
%Y A111672 Cf. A008277, A000326, A005945, A000110, A000258, A000307, A000357, A000405, A111669, A081624.
%Y A111672 Adjacent sequences: A111669 A111670 A111671 this_sequence A111673 A111674 A111675
%Y A111672 Sequence in context: A094954 A083064 A112338 this_sequence A128198 A123349 A123352
%K A111672 nonn,tabl
%O A111672 1,5
%A A111672 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 14 2005

%I A128198
%S A128198 1,1,1,1,2,1,1,3,5,1,1,4,13,16,1,1,5,25,73,65,1,1,6,41,202,527,326,1,1,
%T A128198 7,61,433,2101,4775,1957,1
%N A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (n*k+1)*(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.
%C A128198 The second row (k=1) is sequence A000522 counting the arrangements of a set with n elements and the third row (k=2) is the sequence A128196. Cf. the scheme of variations A128195.
%H A128198 P. Luschny, <a href="http://www.luschny.de/math/seq/variations.html">Variants of Variations</a>.
%e A128198 Array begins:
%e A128198 [k=0] 1, 1, 1, 1, 1, 1, 1, 1
%e A128198 [k=1] 1, 2, 5, 16, 65, 326, 1957, 13700
%e A128198 [k=2] 1, 3, 13, 73, 527, 4775, 52589, 683785
%e A128198 [k=3] 1, 4, 25, 202, 2101, 27556, 441625, 8393062
%e A128198 [k=4] 1, 5, 41, 433, 5885, 101069, 2126545, 53180009
%e A128198 [k=5] 1, 6, 61, 796, 13361, 283706, 7391981, 229229536
%e A128198 [k=6] 1, 7, 85, 1321, 26395, 667651, 20743837, 767801905
%e A128198 [k=7] 1, 8, 113, 2038, 47237, 1386680, 50038129, 2152463090
%p A128198 VarScheme := (k,n) -> `if`(n=0,1,(n*k+1)*(VarScheme(k,n-1)+k^n)); ArrScheme := (k,n) -> `if`(n=0,1, VarScheme(k,n-1)+k^n);
%Y A128198 Cf. A000522, A128195, A128196, A126062.
%Y A128198 Adjacent sequences: A128195 A128196 A128197 this_sequence A128199 A128200 A128201
%Y A128198 Sequence in context: A083064 A112338 A111672 this_sequence A123349 A123352 A114163
%K A128198 easy,nonn,tabl
%O A128198 0,5
%A A128198 Peter Luschny (peter(AT)luschny.de), Mar 02 2007

%I A123349
%S A123349 1,1,1,1,2,1,1,3,5,1,1,4,14,10,1,1,5,30,46,17,1,1,6,55,146,117,26,1,1,7,
%T A123349 91,371,517,251,37,1,1,8,140,812,1742,1476,478,50,1,1,9,204,1596,4878,
%U A123349 6376,3614,834,65,1,1,10,285,2892,11934,22252,19490,7890,1361,82,1,1,11
%N A123349 Square array of Kekule numbers for the mirror-symmetrical chevrons Ch(m,n), read by antidiagonals (m,n>=0).
%C A123349 T(m,1)=A002522(m); T(m,2)=A123350(m); T(m,3)=A123351(m).
%D A123349 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 119-120).
%F A123349 T(m,n)=Sum((binom(m+i-1,i))^2, i=0..n).
%e A123349 T(1,1)=2 because Ch(1,1) consists of a single hexagon; it has 2 perfect matchings: {1,3,5} and {2,4,6}, the edges of the hexagon being labeled consecutively by 1,2,3,4,5,6.
%e A123349 Square array starts:
%e A123349 1 1 1 1 1 1 1 1...
%e A123349 1 2 3 4 5 6 7 8...
%e A123349 1 5 14 30 55 91 140 204 ...
%e A123349 1 10 46 146 371 812 1596 2892 ...
%e A123349 1 17 117 517 1742 4878 11934 26334 ...
%p A123349 T:=(m,n)->sum(binomial(m+i-1,i)^2,i=0..n): TT:=(m,n)->T(m-1,n-1): matrix(9,9,TT); # yields sequence in matrix form
%Y A123349 Cf. A002522, A123350, A123351.
%Y A123349 Adjacent sequences: A123346 A123347 A123348 this_sequence A123350 A123351 A123352
%Y A123349 Sequence in context: A112338 A111672 A128198 this_sequence A123352 A114163 A090234
%K A123349 nonn,tabl
%O A123349 0,5
%A A123349 njas, Oct 14 2006
%E A123349 Edited by Emeric Deutsch, Oct 27 2006, Oct 28 2006

%I A123352
%S A123352 1,1,1,1,2,1,1,3,5,1,1,4,14,14,1,1,5,30,84,42,1,1,6,55,330,594,132,1,1,
%T A123352 7,91,1001,4719,4719,429,1,1,8,140,2548,26026,81796,40898,1430,1,1,9,
%U A123352 204,5712,111384,884884,1643356,379236,4862,1
%N A123352 Triangle read by rows, giving numbers of benzenoids (see the Cyvin-Gutman book for details).
%C A123352 There is another version in A078920 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 12 2007
%D A123352 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
%e A123352 Triangle begins:
%e A123352 1
%e A123352 1 1
%e A123352 1 2 1
%e A123352 1 3 5 1
%e A123352 1 4 14 14 1
%e A123352 1 5 30 84 42 1
%e A123352 1 6 55 330 594 132 1
%e A123352 1 7 91 1001 4719 4719 429 1
%Y A123352 Diagonals give A000108, A005700, A006149, A006150, A006151, etc.
%Y A123352 Cf. A000012, A000027, A000330, A006858, A091962.
%Y A123352 Adjacent sequences: A123349 A123350 A123351 this_sequence A123353 A123354 A123355
%Y A123352 Sequence in context: A111672 A128198 A123349 this_sequence A114163 A090234 A007754
%K A123352 nonn,tabl
%O A123352 0,5
%A A123352 njas, Oct 14 2006
%E A123352 More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 12 2007

%I A114163
%S A114163 1,1,1,1,2,1,1,3,5,1,1,4,18,10,1,1,5,58,68,21,1,1,6,179,398,299,42,1,1,
%T A114163 7,543,2169,3687,1181,85,1,1,8,1636,11388,42726,28488,4836,170,1,1,9,
%U A114163 4916,58576,481374,640974,236436,19286,341,1,1,10,14757,297796,5353690
%N A114163 Triangle read by rows, based on a simple Jacobsthal number recursion rule.
%C A114163 Subdiagonal S(n+1,n) is A000975(n+1). Row sums of inverse are 0^n.
%F A114163 Number triangle T(n, k)=T(n-1, k-1)+J(k+1)*T(n-1, k) where J(n)=A001045(n); Column k has g.f. x^k/Product(1-J(i+1)x, i, 0, k).
%e A114163 Triangle begins
%e A114163 1....1....3....5...11...21...43....J(k+1)
%e A114163 1
%e A114163 1....1
%e A114163 1....2....1
%e A114163 1....3....5....1
%e A114163 1....4...18...10....1
%e A114163 1....5...58...68...21....1
%e A114163 1....6..179..398..299...42....1
%e A114163 For example, T(6,3)=398=58+5*68=T(5,2)+J(4)*T(5,3).
%Y A114163 Cf. A111669.
%Y A114163 Adjacent sequences: A114160 A114161 A114162 this_sequence A114164 A114165 A114166
%Y A114163 Sequence in context: A128198 A123349 A123352 this_sequence A090234 A007754 A058732
%K A114163 easy,nonn,tabl
%O A114163 0,5
%A A114163 Paul Barry (pbarry(AT)wit.ie), Nov 14 2005

%I A090234
%S A090234 2,1,1,3,5,1,3,1,5,11,15,1,1,21,5,11,3,1,43,9,17,13,1,1,9,7,41,23,21,25,
%T A090234 27,9,3,1,9,3,13,11,19,3,1,35,1,23,29,5,11,27,1,39,15,9,47,75,73,11,61,
%U A090234 1,57,3,93,57,117,69,7,93,51,33,97,143,87,65,261,15,23,39,29,55,45,47
%N A090234 Binomial transform gives a prime.
%e A090234 For n = 3 we have (2,1,1,3).(1,3,3,1) = 2*1 + 1*3 + 1*3 + 3*1 = 11 is a prime.
%p A090234 a:=[]: for n from 0 to 100 do m:=add(a[i+1]*binomial(n,i),i=0..n-1): a:=[op(a),nextprime(m)-m] od: op(a); (Ronaldo)
%Y A090234 Cf. A090235.
%Y A090234 Adjacent sequences: A090231 A090232 A090233 this_sequence A090235 A090236 A090237
%Y A090234 Sequence in context: A123349 A123352 A114163 this_sequence A007754 A058732 A060082
%K A090234 nonn
%O A090234 0,1
%A A090234 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 26 2003
%E A090234 More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 26 2004

%I A007754
%S A007754 1,1,1,1,2,1,1,3,5,2,1,4,11,18,7,1,5,19,52,85,33,1,6,29,110,301,492,
%T A007754 191,1,7,41,198,751,2055,3359,1304,1,8,55,322,1555,5898,16139,26380,
%U A007754 10241,1,9,71,488,2857,13797,52331,143196,234061,90865,1,10,89,702
%N A007754 Array (a frieze pattern) defined by a(n,k) = (a(n-1,k)*a(n-1,k+1) - 1) / a(n-2,k+1), read by antidiagonals.
%D A007754 Email from James Propp (propp(AT)math.wisc.edu), Nov. 28, 2000.
%F A007754 a(n, k)=(n+k)*a(n-1, k)-a(n-2, k) with a(0, k)=1 and a(-1, k)=0 - Henry Bottomley (se16(AT)btinternet.com), Feb 28 2001
%F A007754 a(n, k) = Pi*(BesselJ(n+k+1, 2)*BesselY(k, 2) - BesselY(n+k+1, 2)*BesselJ(k, 2)) - Alec Mihailovs (alec(AT)mihailovs.com), Aug 21 2005
%F A007754 Column asymptotics (i.e. for fixed k and n -> infinity): a(n, k) ~ BesselJ(k, 2)*(n+k)! - Alec Mihailovs (alec(AT)mihailovs.com), Aug 21 2005
%e A007754 Array begins:
%e A007754 1 1 1 1 1 1 1 1 ...
%e A007754 1 2 3 4 5 6 7 ...
%e A007754 1 5 11 19 29 41 ...
%e A007754 2 18 52 110 198 ...
%e A007754 7 85 301 751 ...
%Y A007754 Row 0-3: A000012, A000027(n+1), A028387, A058794-A058796. Columns 0-2: A058797-A058799.
%Y A007754 Adjacent sequences: A007751 A007752 A007753 this_sequence A007755 A007756 A007757
%Y A007754 Sequence in context: A123352 A114163 A090234 this_sequence A058732 A060082 A102225
%K A007754 nonn,easy,nice,tabl
%O A007754 0,5
%A A007754 njas, Nov 28 2000
%E A007754 More terms from Christian G. Bower (bowerc(AT)usa.net), Dec 02 2000

%I A058732
%S A058732 1,0,2,1,1,3,5,2,5,3,7,7,12
%N A058732 McKay-Thompson series of class 60b for Monster.
%D A058732 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058732 T60b = 1/q + 2*q^3 + q^5 + q^7 + 3*q^9 + 5*q^11 + 2*q^13 + 5*q^15 + 3*q^17 + ...
%Y A058732 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058732 Adjacent sequences: A058729 A058730 A058731 this_sequence A058733 A058734 A058735
%Y A058732 Sequence in context: A114163 A090234 A007754 this_sequence A060082 A102225 A075248
%K A058732 nonn
%O A058732 -1,3
%A A058732 njas, Nov 27, 2000

%I A060082
%S A060082 1,1,1,1,2,1,1,3,5,3,1,4,14,28,17,1,5,30,126,255,155,1,6,55,396,1683,3410,
%T A060082 2073,1,7,91,1001,7293,31031,62881,38227,1,8,140,2184,24310,177320,754572,
%U A060082 1529080,929569,1,9,204,4284,67626,753610,5497596,23394924,47408019
%V A060082 1,1,-1,1,-2,1,1,-3,5,-3,1,-4,14,-28,17,1,-5,30,-126,255,-155,1,-6,55,-396,1683,-3410,
%W A060082 2073,1,-7,91,-1001,7293,-31031,62881,-38227,1,-8,140,-2184,24310,-177320,754572,
%X A060082 -1529080,929569,1,-9,204,-4284,67626,-753610,5497596,-23394924,47408019
%N A060082 Coefficients of even indexed Euler polynomials (falling powers without zeros).
%C A060082 E(2n,x) = x^(2n) + Sum[k=1..n, a(n,k)*x^(2n-2k+1) ].
%D A060082 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
%H A060082 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%H A060082 Z.-W. Sun, <a href="http://pweb.nju.edu.cn/zwsun/papers.htm">Introduction to Bernoulli and Euler polynomials</a>
%F A060082 E(n, x) = 2/(n+1) * [B(n+1, x) - 2^(n+1)*B(n+1, x/2) ], with B(n, x) the Bernoulli polynomials.
%e A060082 E(0,x) = 1.
%e A060082 E(2,x) = x^2 - x.
%e A060082 E(4,x) = x^4 - 2*x^3 + x.
%e A060082 E(6,x) = x^6 - 3*x^5 + 5*x^3 - 3*x.
%e A060082 E(8,x) = x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x.
%e A060082 E(10,x) = x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x.
%o A060082 (PARI) {B(n,v='x)=sum(i=0,n,binomial(n,i)*bernfrac(i)*v^(n-i))} E(n,v='x)=2/(n+1)*(B(n+1,v)-2^(n+1)*B(n+1,v/2)) /* from R. Stephan */
%Y A060082 E(2n, 1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
%Y A060082 -E(2n, -1/2)*(-4)^n/3 = A076552(n), -E(2n, 1/3)*(-9)^n/2 = A002114(n).
%Y A060082 Cf. A060083 (rising powers), A060096-7 (Euler polynomials), A004172 (with zeros).
%Y A060082 Columns (left edge) include A000330, A053132. Columns (right edge) include A001469.
%Y A060082 Adjacent sequences: A060079 A060080 A060081 this_sequence A060083 A060084 A060085
%Y A060082 Sequence in context: A090234 A007754 A058732 this_sequence A102225 A075248 A128325
%K A060082 sign,easy,tabl
%O A060082 0,5
%A A060082 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 29 2001
%E A060082 Edited by Ralf Stephan, Nov 05 2004

%I A102225
%S A102225 1,1,1,2,1,1,3,5,3,1,7,20,19,5,1,17,51,57,33,11,1,75,289,438,345,143,21,1,346,
%T A102225 1426,2441,2073,767,97,43,1,4874,22622,44289,46156,26231,7713,1183,85,1,49047,
%U A102225 259734,530214,520395,272461,103617,30735,1119,171,1,3009094,15968025,35495592
%V A102225 1,1,1,2,-1,1,3,-5,3,1,7,-20,19,-5,1,17,-51,57,-33,11,1,75,-289,438,-345,143,-21,1,346,
%W A102225 -1426,2441,-2073,767,-97,43,1,4874,-22622,44289,-46156,26231,-7713,1183,-85,1,49047,
%X A102225 -259734,530214,-520395,272461,-103617,30735,1119,171,1,3009094,-15968025,35495592
%N A102225 Triangular matrix, read by rows, where row k is formed from the first differences of row (k-1) of its matrix square, with an appended '1' for the main diagonal.
%C A102225 Row sums are: {1,2,2,2,2,2,...}. Column 0 is A102226. Column 1 is A102227. Matrix square forms A102228.
%F A102225 T(n, k) = [T^2](n-1, k) - [T^2](n-1, k-1) for n>k>0, with T(n, n)=1 for n>=0, and T(n, 0) = [T^2](n-1, 0) for n>0.
%e A102225 Rows begin:
%e A102225 [1],
%e A102225 [1,1],
%e A102225 [2,-1,1],
%e A102225 [3,-5,3,1],
%e A102225 [7,-20,19,-5,1],
%e A102225 [17,-51,57,-33,11,1],
%e A102225 [75,-289,438,-345,143,-21,1],
%e A102225 [346,-1426,2441,-2073,767,-97,43,1],
%e A102225 [4874,-22622,44289,-46156,26231,-7713,1183,-85,1],...
%e A102225 The matrix square A102225^2 forms A102228:
%e A102225 [1],
%e A102225 [2,1],
%e A102225 [3,-2,1],
%e A102225 [7,-13,6,1],
%e A102225 [17,-34,23,-10,1],
%e A102225 [75,-214,224,-121,22,1],...
%e A102225 The first differences of the rows of A102228
%e A102225 form A102225 excluding the main diagonal of 1's.
%o A102225 (PARI) {T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,1]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]-(A^2)[i-1,j-1]));));A=B);return(A[n+1,k+1])}
%Y A102225 Cf. A102226, A102227, A102228.
%Y A102225 Adjacent sequences: A102222 A102223 A102224 this_sequence A102226 A102227 A102228
%Y A102225 Sequence in context: A007754 A058732 A060082 this_sequence A075248 A128325 A111528
%K A102225 sign,tabl
%O A102225 0,4
%A A102225 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 01 2005

%I A075248
%S A075248 0,1,2,1,1,3,5,9,6,3,12,5,18,15,10,5,21,11,22,18,15,8,55,30,15,20,43,
%T A075248 20,45,5,24,35,23,36,53,10,21,52,62,6,62,12,73,69,16,11,92,38,84,34,50,
%U A075248 11,77,56,80,45,38,34,142,6,23,96,53,53,67,15,66,70,124,12,148,21,57
%N A075248 Number of solutions (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.
%C A075248 All of the solutions can be printed by removing the comment symbols from the Mathematica program. For the solution (x,y,z) having the largest z value, see (A075249, A075250, A075251). See A073101 for the 4/n conjecture due to Erdos and Straus.
%H A075248 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b075248.txt">Table of n, a(n) for n=2..1000</a>
%H A075248 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A075248 Ron Knott <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fractions/egyptian.html">Egyptian Fractions</a>
%e A075248 a(4)=2 because there are two solutions: 5/4 = 1/1+1/5+1/20 and 5/4 = 1/1+1/6+1/12.
%t A075248 m = 5; For[lst = {}; n = 2, n <= 100, n++, cnt = 0; xr = n/m; If[IntegerQ[xr], xMin = xr + 1, xMin = Ceiling[xr]]; If[IntegerQ[3xr], xMax = 3xr - 1, xMax = Floor[3xr]]; For[x = xMin, x <= xMax, x++, yr = 1/(m/n - 1/x); If[IntegerQ[yr], yMin = yr + 1, yMin = Ceiling[yr]]; If[IntegerQ[2yr], yMax = 2yr + 1, yMax = Ceiling[2yr]]; For[y = yMin, y <= yMax, y++, zr = 1/(m/n - 1/x - 1/y); If[y > x && zr > y && IntegerQ[zr], z = zr; cnt++; (*Print[n, " ", x, " ", y, " ", z]*)]]]AppendTo[lst, cnt]]; lst
%Y A075248 Cf. A073101, A075249, A075250, A075251.
%Y A075248 Adjacent sequences: A075245 A075246 A075247 this_sequence A075249 A075250 A075251
%Y A075248 Sequence in context: A058732 A060082 A102225 this_sequence A128325 A111528 A107702
%K A075248 nice,nonn
%O A075248 2,3
%A A075248 T. D. Noe (noe(AT)sspectra.com), Sep 10 2002

%I A128325
%S A128325 1,1,1,1,1,2,1,1,3,6,1,1,4,12,23,1,1,5,20,57,104,1,1,6,30,114,305,531,1,
%T A128325 1,7,42,200,712,1787,2982,1,1,8,56,321,1435,4772,11269,18109,1,1,9,72,
%U A128325 483,2608,10900,33896,75629,117545,1,1,10,90,692,4389,22219,86799
%N A128325 Rectangular table, read by antidiagonals, where the g.f.s of row n, R(x,n), satisfy: R(x,n+1) = R(G(x),n) for n>=0, and x*R(x,0) = G(x) = x + x*G(G(x)) is the g.f. of A030266.
%C A128325 Row n equals 1 + (n+2)-th self-composition of the g.f. G(x) of A030266: R(x,0) = 1 + G(G(x); R(x,1) = 1 + G(G(G(x))); R(x,2) = 1 + G(G(G(G(x)))); etc.
%F A128325 R(x,n) = 1 + x*Product_{k=0..n+1} R(x,k) . R(x,n) = 1 + x/[1 - x*Sum_{k=1..n+2} R(x,k) ].
%e A128325 Consider the infinite system of simultaneous equations:
%e A128325 A = 1 + xAB;
%e A128325 B = 1 + xABC;
%e A128325 C = 1 + xABCD;
%e A128325 D = 1 + xABCDE;
%e A128325 E = 1 + xABCDEF; ...
%e A128325 The unique solution to the variables are:
%e A128325 A = R(x,0), B = R(x,1), C = R(x,2), D = R(x,3), E = R(x,4), etc.,
%e A128325 where R(x,n) denotes the g.f. of row n of this table and satisfies:
%e A128325 R(x,1) = R(x*A,0); R(x,2) = R(x*A,1); R(x,3) = R(x*A,2); etc.
%e A128325 The row g.f.s are also related by:
%e A128325 R(x,0) = 1 + x/(1 - x*R(x,1) - x*R(x,2));
%e A128325 R(x,1) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3));
%e A128325 R(x,2) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3) - x*R(x,4)); etc.
%e A128325 The initial rows of this table begin:
%e A128325 R(x,0):[1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764, ...];
%e A128325 R(x,1):[1, 1, 3, 12, 57, 305, 1787, 11269, 75629, 535960, 3987913,...];
%e A128325 R(x,2):[1, 1, 4, 20, 114, 712, 4772, 33896, 253102, 1975610, ...];
%e A128325 R(x,3):[1, 1, 5, 30, 200, 1435, 10900, 86799, 720074, 6196295, ...];
%e A128325 R(x,4):[1, 1, 6, 42, 321, 2608, 22219, 196910, 1805899, 17079151, ...];
%e A128325 R(x,5):[1, 1, 7, 56, 483, 4389, 41531, 406441, 4095749, 42371678, ...];
%e A128325 R(x,6):[1, 1, 8, 72, 692, 6960, 72512, 777888, 8559852, 96364708, ..];
%e A128325 R(x,7):[1, 1, 9, 90, 954, 10527, 119832, 1399755, 16720998, ...];
%e A128325 R(x,8):[1, 1, 10, 110, 1275, 15320, 189275, 2392998, 30865353, ...];
%e A128325 R(x,9):[1, 1, 11, 132, 1661, 21593, 287859, 3918189, 54301621, ...];
%e A128325 R(x,10):[1, 1, 12, 156, 2118, 29624, 423956, 6183400, 91673594, ...].
%o A128325 (PARI) {T(n,k)=local(A=vector(n+k+3,m,1+x+x*O(x^(n+k)))); for(i=1,n+k+3,for(j=1,n+k+1,N=n+k+2-j; A[N]=1+x/(1-x*sum(m=2,N+2,A[m]+x*O(x^(n+k))))));Vec(A[n+1])[k+1]}
%Y A128325 Cf. A030266 (row 0), A128326 (row 1), A128327 (row 2), A128328 (row 3), A128329 (main diagonal); A128330 (variant).
%Y A128325 Adjacent sequences: A128322 A128323 A128324 this_sequence A128326 A128327 A128328
%Y A128325 Sequence in context: A060082 A102225 A075248 this_sequence A111528 A107702 A111670
%K A128325 nonn,tabl
%O A128325 0,6
%A A128325 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2007

%I A111528
%S A111528 1,1,1,1,1,2,1,1,3,6,1,1,4,13,24,1,1,5,22,71,120,1,1,6,33,148,461,720,1,
%T A111528 1,7,46,261,1156,3447,5040,1,1,8,61,416,2361,10192,29093,40320,1,1,9,78,
%U A111528 619,4256,23805,99688,273343,362880,1,1,10,97,876,7045,48096,263313
%N A111528 Square table, read by antidiagonals, where the g.f. for row n+1 is generated by: R_{n+1}(x) = (1+n*x - 1/R_n(x))/(n+1) with R_0(x) = Sum_{n>=0} n!*x^n.
%F A111528 T(n, 0) = 1, T(0, k) = k!, else for n>=1 and k>=1: T(n, k) = (T(n-1, k+1)-T(n-1, k))/n - Sum_{j=1..k-1} T(n, j)*T(n-1, k-j); also T(n, k) = (k/n)*[x^k] Log[ Sum_{m=0..k} (n-1+m)!/(n-1)!*x^m)].
%F A111528 T(n, k) = Sum_{j = 0...k} A089949(k, j)*n^(k-j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 08 2005
%F A111528 R_n(x) = -(n-1)!/n/Sum((i+n-2)!*x^i,i = 1 .. infinity), n>0. - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 06 2006
%e A111528 Table begins:
%e A111528 1,1,2,6,24,120,720,5040,40320,...
%e A111528 1,1,3,13,71,461,3447,29093,273343,...
%e A111528 1,1,4,22,148,1156,10192,99688,1069168,...
%e A111528 1,1,5,33,261,2361,23805,263313,3161781,...
%e A111528 1,1,6,46,416,4256,48096,591536,7840576,...
%e A111528 1,1,7,61,619,7045,87955,1187845,17192275,...
%e A111528 1,1,8,78,876,10956,149472,2195208,34398288,...
%e A111528 1,1,9,97,1193,16241,240057,3804353,64092553,...
%e A111528 1,1,10,118,1576,23176,368560,6262768,112784896,...
%e A111528 Rows are generated by logarithms of factorial series:
%e A111528 Log(1 + x + 2*x^2 + 6*x^3 + 24*x^4 +... n!*x^n +...) = x + 3/2*x^2 + 13/3*x^3 + 71/4*x^4 + 461/5*x^5 +...
%e A111528 (1/2)*Log(1 + 2*x + 6*x^2 +... + (n+1)!/1!*x^n +...) = x + 4/2*x^2 + 22/3*x^3 + 148/4*x^4 + 1156/5*x^5 +...
%e A111528 (1/3)*Log(1 + 3*x + 12*x^2 +60*x^3+... +(n+2)!/2!*x^n +...) = x + 5/2*x^2 + 33/3*x^3 + 261/4*x^4 + 2361/5*x^5 +...
%e A111528 G.f. of row n may be expressed by the continued fraction:
%e A111528 R_n(x) = 1/(1+n*x - (n+1)*x/(1+(n+1)*x - (n+2)*x/(1+(n+2)*x -...
%e A111528 or recursively by: R_n(x) = 1/(1+n*x - (n+1)*R_{n+1}(x)).
%o A111528 (PARI) {T(n,k)=if(n<0|k<0,0,if(k==0|k==1,1,if(n==0,k!, (T(n-1,k+1)-T(n-1,k))/n-sum(j=1,k-1,T(n,j)*T(n-1,k-j)))))} (PARI) {T(n,k)=if(n<0|k<0,0,if(k==0,1,if(n==0,k!, k/n*polcoeff(log(sum(m=0,k,(n-1+m)!/(n-1)!*x^m)),k))))}
%Y A111528 Cf: A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111533 (row 6), A111534 (diagonal).
%Y A111528 Adjacent sequences: A111525 A111526 A111527 this_sequence A111529 A111530 A111531
%Y A111528 Sequence in context: A102225 A075248 A128325 this_sequence A107702 A111670 A123353
%K A111528 nonn,tabl
%O A111528 0,6
%A A111528 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 06 2005

%I A107702
%S A107702 1,1,1,1,2,1,1,3,6,1,1,4,15,22,1,1,5,28,93,90,1,1,6,45,244,645,394,1,1,
%T A107702 7,66,505,2380,4791,1806,1,1,8,91,906,6345,24868,37275,8558,1,1,9,120,
%U A107702 1477,13926,85405,272188,299865,41586,1,1,10,153,2248,26845,229326
%N A107702 Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.
%C A107702 Row sums are A107703. Transpose of square array A103209, read by anti-diagonals.
%F A107702 Number triangle T(n, k)=if(k<=n, sum{j=0..k, C(k+j, 2j)(n-k)^j*C(j)}, 0), C(n) given by A000108.
%Y A107702 Adjacent sequences: A107699 A107700 A107701 this_sequence A107703 A107704 A107705
%Y A107702 Sequence in context: A075248 A128325 A111528 this_sequence A111670 A123353 A056646
%K A107702 easy,nonn,tabl
%O A107702 0,5
%A A107702 Paul Barry (pbarry(AT)wit.ie), May 21 2005

%I A111670
%S A111670 1,1,1,1,2,1,1,3,6,1,1,4,15,24,1,1,5,28,105,116,1,1,6,45,280,929,648,1,
%T A111670 1,7,66,585,3600,9851,4088,1,1,8,91,1056,9865,56240,121071,28640,1
%N A111670 Triangle, generated from A039755.
%C A111670 First few rows of the array generated from A039775 are: 1, 1, 1, 1, 1, 1,... 1, 2, 6, 24, 116, 648,... 1, 3, 15, 105, 929,... 1, 4, 28, 280, 3600,... 1, 5, 45, 585, 9865,... Column 2, (1, 6, 15, 28...) = A000384; column 3 (1, 24, 105...) = A011199; antidiagonal 1, 2, 6, 24, 116...of the triangle = A007405.
%F A111670 Let A039755 (an analogue of Stirling numbers of the second kind) = an infinite lower triangular matrix M; then generate an array using M^n * [1, 0, 0, 0...]. Take antidiagonals of the array which form the rows of a new triangle A111670.
%e A111670 The antidiagonal (1, 4, 15, 24, 1) of the array becomes row 4 of the triangle.
%Y A111670 Cf. A039755, A007405, A000384, A011199.
%Y A111670 Adjacent sequences: A111667 A111668 A111669 this_sequence A111671 A111672 A111673
%Y A111670 Sequence in context: A128325 A111528 A107702 this_sequence A123353 A056646 A056056
%K A111670 nonn,tabl
%O A111670 0,5
%A A111670 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 14 2005

%I A123353
%S A123353 1,1,1,1,2,1,1,3,6,1,1,4,20,19,1,1,5,50,155,62,1,1,6,105,805,1315,
%T A123353 207,1
%N A123353 Triangle read by rows, giving numbers of benzenoids (see the Cyvin-Gutman book for details).
%D A123353 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 188).
%e A123353 Triangle begins:
%e A123353 1
%e A123353 1 1
%e A123353 1 2 1
%e A123353 1 3 6 1
%e A123353 1 4 20 19 1
%e A123353 1 5 50 155 62 1
%e A123353 1 6 105 805 1315 207 1
%Y A123353 Diagonals give A026012, A123355, A123356.
%Y A123353 Adjacent sequences: A123350 A123351 A123352 this_sequence A123354 A123355 A123356
%Y A123353 Sequence in context: A111528 A107702 A111670 this_sequence A056646 A056056 A136462
%K A123353 nonn,tabl,more
%O A123353 0,5
%A A123353 njas, Oct 14 2006

%I A056646
%S A056646 1,1,1,1,1,2,1,1,3,6,1,2,2,1,3,3,1,2,1,2,2,1,1,2,10,5,10,5,3,12,3,3,1,
%T A056646 2,5,10,10,5,3,6,2,1,2,1,5,20,15,30,42,21,14,7,7,28,2,1,1,4,1,4,4,2,21,
%U A056646 21,7,14,7,14,6,3,1,2,2,1,10,5,35,140,7,14,126,63,2,1,5,20,90,45,3,12
%N A056646 Square root of largest unitary square divisor of central binomial coefficient.
%F A056646 a(n)=A000188[A001405(n)]/A055229[A001405(n)]= A056056(n)/A056059(n)
%e A056646 n=28, binomial[28,14]=2.2.2.3.3.3.5.5.17.19.23. The square root of its largest square divisor is 30, while the largest unitary analogue is a(28)=30/6=5.
%Y A056646 A000188, A001405, A055229, A056056, A056059.
%Y A056646 Adjacent sequences: A056643 A056644 A056645 this_sequence A056647 A056648 A056649
%Y A056646 Sequence in context: A107702 A111670 A123353 this_sequence A056056 A136462 A060517
%K A056646 nonn
%O A056646 1,6
%A A056646 Labos E. (labos(AT)ana.sote.hu), Aug 09 2000

%I A056056
%S A056056 1,1,1,1,1,2,1,1,3,6,1,2,2,2,3,3,1,2,1,2,2,2,1,2,10,10,30,30,6,12,3,3,
%T A056056 3,6,5,10,10,10,3,6,2,2,2,2,30,60,15,30,42,42,42,42,14,28,2,2,2,4,2,4,
%U A056056 4,4,21,21,7,14,7,14,6,6,1,2,2,2,10,10,70,140,7,14,126,126,6,6,30,60
%N A056056 Square root of largest square dividing n-th central binomial coefficient.
%F A056056 a(n)=A000188[A001405(n)]
%Y A056056 Cf. A001405, A000188, A008833, A007913, A055229, A055231, A056057-A056061.
%Y A056056 Adjacent sequences: A056053 A056054 A056055 this_sequence A056057 A056058 A056059
%Y A056056 Sequence in context: A111670 A123353 A056646 this_sequence A136462 A060517 A074662
%K A056056 nonn
%O A056056 1,6
%A A056056 Labos E. (labos(AT)ana.sote.hu), Jul 26 2000

%I A136462
%S A136462 1,1,1,1,2,1,1,3,6,4,1,4,15,56,70,1,5,28,220,1820,4368,1,6,45,560,10626,
%T A136462 201376,906192,1,7,66,1140,35960,1712304,74974368,621216192,1,8,91,2024,
%U A136462 91390,7624512,927048304,94525795200,1429702652400,1,9,120,3276,194580
%N A136462 Square table, read by antidiagonals, where T(n,k) = C((n+1)*2^(k-1), k) for n>=0, k>=0.
%C A136462 Row n equals column 0 of matrix product A136467^(n+1) for n>=0.
%F A136462 O.g.f. of row n: Sum_{k>=0} ((n+1)/2)^k*log(1 + 2^k*x)^k/k! = Sum_{k>=0} C((n+1)*2^(k-1), k)*x^k for n>=0.
%e A136462 1,1,1,4,70,4368,906192,621216192,1429702652400,11288510714272000,...;
%e A136462 1,2,6,56,1820,201376,74974368,94525795200,409663695276000,...;
%e A136462 1,3,15,220,10626,1712304,927048304,1708566412608,...;
%e A136462 1,4,28,560,35960,7624512,5423611200,13161885792000,...;
%e A136462 1,5,45,1140,91390,24040016,21193254160,63815149590720,...;
%e A136462 1,6,66,2024,194580,61124064,64300886496,231207760388736,...;
%e A136462 1,7,91,3276,367290,134153712,163995687856,685581099291712,...;
%e A136462 1,8,120,4960,635376,264566400,368532802176,1756185841659392,...; ...
%e A136462 Triangle A136467 begins:
%e A136462 1;
%e A136462 1,1;
%e A136462 1,4,1;
%e A136462 4,32,16,1;
%e A136462 70,848,576,64,1;
%e A136462 4368,75648,62208,9216,256,1;
%e A136462 906192,22313216,21169152,3792896,143360,1024,1;
%e A136462 621216192,21827627008,23212261376,4793434112,223215616,2228224,4096,1;
%e A136462 such that row n of A136462 equals column 0 of A136467^(n+1).
%o A136462 (PARI) {T(n,k)=binomial((n+1)*2^(k-1),k)} (PARI) /* T(n,k) = Coefficient of x^k in series: */ {T(n,k)=polcoeff(sum(i=0,k,((n+1)/2)^i*log(1+2^i*x +x*O(x^k))^i/i!),k)}
%Y A136462 Cf. rows: A136465, A014070, A136466, A101346; A136463 (diagonal); A136467.
%Y A136462 Adjacent sequences: A136459 A136460 A136461 this_sequence A136463 A136464 A136465
%Y A136462 Sequence in context: A123353 A056646 A056056 this_sequence A060517 A074662 A025243
%K A136462 nonn,tabl
%O A136462 0,5
%A A136462 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 31 2007

%I A060517
%S A060517 1,1,0,1,1,2,1,1,3,6,6,6,3,1,1,6,15,34,58,60,60,50,33,10,1,1,10,35,120,
%T A060517 265,475,820,1200,1615,1860,1693,1060,425,105,15,1,1,15,75,330,990,
%U A060517 2691,6326,13170,26205,48055,79206,112863,133535,124680,88890,47874
%N A060517 Triangle T(n,k) of series-reduced (or homeomorphically irreducible) graphs with loops on n labeled nodes and with k edges, k=0..binomial(n+1,2).
%D A060517 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%F A060517 E.g.f.: (1 + x * y)^( - 1/2) * exp( - x * y/2 - x^2 * y^2/4) * Sum_{k=0..inf}(1 + x)^binomial(k + 1, 2) * exp( - x^2 * y * k^2/(2 * (1 + x * y)) + x^2 * y * k/2) * x^k/k!
%e A060517 [1], [1, 0], [1, 1, 2, 1], [1, 3, 6, 6, 6, 3, 1], [1, 6, 15, 34, 58, 60, 60, 50, 33, 10, 1], [1, 10, 35, 120, 265, 475, 820, 1200, 1615, 1860, 1693, 1060, 425, 105, 15, 1], [1, 15, 75, 330, 990, 2691, 6326, 13170, 26205, 48055, 79206, 112863, 133535, 124680, 88890, 47874, 19443, 5925, 1330, 210, 21, 1], ...
%Y A060517 Row sums: A060516, A003514, A060514.
%Y A060517 Adjacent sequences: A060514 A060515 A060516 this_sequence A060518 A060519 A060520
%Y A060517 Sequence in context: A056646 A056056 A136462 this_sequence A074662 A025243 A135701
%K A060517 easy,nonn
%O A060517 0,6
%A A060517 Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 24 2001

%I A074662
%S A074662 2,1,1,3,6,8,12,21,35,55,88,144,234,377,609,987,1598,2584,4180,6765,
%T A074662 10947,17711,28656,46368,75026,121393,196417,317811,514230,832040,
%U A074662 1346268,2178309,3524579,5702887,9227464,14930352,24157818,39088169
%N A074662 a(n) = F(n+1)+cos(n*pi/2).
%C A074662 a(n) is the convolution of L(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594. a(2n+1)=F(2n+2), F = Fibonacci numbers.
%C A074662 a(n)=Sum((-1)^(i+Floor(n/2))L(2i+e),(i=0,..,Floor(n/2))), where L(n) Lucas numbers and e=(1/2)(1-(-1)^n).
%F A074662 a(n)=a(n-1)+a(n-3)+a(n-4), a(0)=2, a(1)=1, a(2)=1, a(3)=3. G.f.: (2 - x)/(1 - x - x^3 - x^4).
%F A074662 a(4n)=F(4n+1)+1, a(4n+2)=F(4n+3)-1.
%t A074662 CoefficientList[Series[(2 - x)/(1 - x - x^3 - x^4), {x, 0, 40}], x]
%o A074662 (PARI) a(n)=if(n<0,0,fibonacci(n+1)+real(I^n))
%Y A074662 Cf. A056594.
%Y A074662 Adjacent sequences: A074659 A074660 A074661 this_sequence A074663 A074664 A074665
%Y A074662 Sequence in context: A056056 A136462 A060517 this_sequence A025243 A135701 A051467
%K A074662 easy,nonn
%O A074662 0,1
%A A074662 Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2002

%I A025243
%S A025243 1,2,1,1,3,6,14,33,79,194,482,1214,3090,7936,20544,53545,140399,370098,
%T A025243 980226,2607242,6961462,18652112,50133616,135140598,365254226,989614976,
%U A025243 2687312752,7312725944,19938170096,54460115308,149007155356,408341969073
%N A025243 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
%C A025243 a(n) = number of Dyck (n-1)-paths that contain no DUDUs and no UUDDs (n>=3). For example, a(5)=3 counts UUUDUDDD, UDUUDUDD, UUDUDDUD. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006
%F A025243 G.f.: (1+x+2*x^2-sqrt(1-2*x-3*x^2+4*x^4))/2 - Michael Somos, Jun 08, 2000.
%o A025243 (PARI) a(n)=polcoeff((x+2*x^2-sqrt(1-2*x-3*x^2+4*x^4+x*O(x^n)))/2,n)
%Y A025243 Adjacent sequences: A025240 A025241 A025242 this_sequence A025244 A025245 A025246
%Y A025243 Sequence in context: A136462 A060517 A074662 this_sequence A135701 A051467 A077385
%K A025243 nonn
%O A025243 1,2
%A A025243 Clark Kimberling (ck6(AT)evansville.edu)

%I A135701
%S A135701 1,1,1,2,1,1,3,7,6,4,3,67,13,96,121,11,116,128,19,594,30,131,897,181,
%T A135701 156,2033,3760,2105,1842,6961,41453,7556,28716,9974,108217,3031,256669,
%U A135701 402707
%N A135701 First differences of indices of A000043.
%C A135701 Also, first differences of A016027.
%C A135701 This sequence is related to the perfect numbers A000396 and the Mersenne primes A000668.
%H A135701 A. Booker, <a href="http://primes.utm.edu/nthprime">The Nth Prime Page"</a>.
%F A135701 a(n) = index of A000043(n+1) - index of A000043(n) = Pi(A000043(n+1)) - Pi(A000043(n)) = A016027(n+1) - A016027(n).
%e A135701 a(16)=11 because A000043(16+1)=2281 is the 339 n-th prime, then A16027(16+1)=339 and A000043(16)=2203 is the 328 n-th prime, then A16027(16)=328 and we can write 339-328=11.
%Y A135701 Cf. A000043, A016027, A000396, A000668.
%Y A135701 Adjacent sequences: A135698 A135699 A135700 this_sequence A135702 A135703 A135704
%Y A135701 Sequence in context: A060517 A074662 A025243 this_sequence A051467 A077385 A066013
%K A135701 easy,nonn
%O A135701 1,4
%A A135701 Omar E. Pol (info(AT)polprimos.com), Mar 02 2008

%I A051467
%S A051467 1,1,2,1,1,3,7,8,1,10,15,1,4,25,18,1,1,5,22,56,91,98,70,32,1,27,78,147,
%T A051467 189,168,102,1,6,105,225,336,357,270,50,1,330,561,693,627,1,7,45,176,
%U A051467 891,1254,1320,605,253,72,1,52,221,2145,2574,858,325,1,8,273,2002,4719
%N A051467 (Terms in A029640)/2.
%Y A051467 A029640, A029635.
%Y A051467 Adjacent sequences: A051464 A051465 A051466 this_sequence A051468 A051469 A051470
%Y A051467 Sequence in context: A074662 A025243 A135701 this_sequence A077385 A066013 A014521
%K A051467 nonn
%O A051467 0,3
%A A051467 James A. Sellers (sellersj(AT)math.psu.edu)

%I A077385
%S A077385 1,1,2,1,1,3,9,3,1,1,4,16,64,16,4,1,1,5,25,125,625,125,25,5,1,1,6,36,
%T A077385 216,1296,7776,1296,216,36,6,1,1,7,49,343,2401,16807,117649,16807,2401,
%U A077385 343,49,7,1,1,8,64,512,4096,32768,262144,2097152,262144,32768,4096,512
%N A077385 Triangle read by rows in which n-th row contains 2n-1 terms starting from n^0 to n^(n-1) in increasing order and then in decreasing order to n^0.
%e A077385 1; 1,2,1; 1,3,9,3,1; 1,4,16,64,16,4,1; ...
%p A077385 A077385 := proc(n,k) if k < n then n^k ; else n^(2*n-k-2) ; fi ; end: for n from 1 to 10 do for k from 0 to 2*n-2 do printf("%d, ",A077385(n,k)) ; od : od : - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 03 2007
%Y A077385 Cf. A077386.
%Y A077385 Adjacent sequences: A077382 A077383 A077384 this_sequence A077386 A077387 A077388
%Y A077385 Sequence in context: A025243 A135701 A051467 this_sequence A066013 A014521 A084389
%K A077385 nonn,tabf
%O A077385 1,3
%A A077385 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 06 2002
%E A077385 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 03 2007

%I A066013
%S A066013 1,1,1,1,2,1,1,3,11,5,3,39,8,1,15
%N A066013 Number of codes having highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105688.
%H A066013 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A066013 S. T. Dougherty, M. Harada and P. Sole', <a href="http://academic.uofs.edu/faculty/Doughertys1/publ.htm">Shadow Codes over Z_4</a>, Finite Fields Applic., 7 (2001), 507-529.
%H A066013 P. Gaborit, <a href="http://www.unilim.fr/pages_perso/philippe.gaborit/SD/">Tables of Self-Dual Codes</a>
%H A066013 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (<a href="http://www.research.att.com/~njas/doc/self.txt">Abstract</a>, <a href="http://www.research.att.com/~njas/doc/self.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/self.ps">ps</a>).
%Y A066013 Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.
%Y A066013 Cf. A066012 for minimal Lee distances of these codes. See also A066014-A066017.
%Y A066013 Adjacent sequences: A066010 A066011 A066012 this_sequence A066014 A066015 A066016
%Y A066013 Sequence in context: A135701 A051467 A077385 this_sequence A014521 A084389 A122022
%K A066013 nonn
%O A066013 1,5
%A A066013 njas, Dec 11 2001; revised May 06 2005

%I A014521
%S A014521 2,1,1,3,12,52,288,1871,14034,119292,1133278,11899423,136843365,
%T A014521 1710542068,23092317922,334838609873,5189998453040,85634974475162,
%U A014521 1498612053315336,27724322986333718,540624298233507504
%N A014521 Nearest integer to GAMMA(n+1/2).
%D A014521 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 255.
%H A014521 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%H A014521 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 255.
%p A014521 [ seq(round(evalf(GAMMA(n+1/2),100)), n=0..24) ];
%t A014521 Table[Round[Gamma[n + 1/2]], {n, 0, 26}]
%Y A014521 Adjacent sequences: A014518 A014519 A014520 this_sequence A014522 A014523 A014524
%Y A014521 Sequence in context: A051467 A077385 A066013 this_sequence A084389 A122022 A049258
%K A014521 nonn
%O A014521 0,1
%A A014521 njas

%I A084389
%S A084389 2,1,1,3,13,1,2,7,5,3,1,2,19,1,4,3,11,6,2,5,1,3,9,2,4,1,8,14,152,5,2,45,
%T A084389 1,10,2,3,18,7,1,5,95,4,15,13,6,9,2,3,53,1,12,17,16,5,4,2,31,7,3,1,6,21,
%U A084389 105,23,10,2,33,5,1,3,9,14,6,7,2,11,4,1,84,19,3,5
%N A084389 Solutions n for n^3 + m = k^2.
%H A084389 Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/n37ltk2.msnw">Proof that n^3+7 <> k^2 for all integers n,k</a>.
%o A084389 (PARI) n3pmsq4(n,m1) = { for(m=1,m1, for(x=1,n, y=x^3+m; if(issquare(y),print1(x","); break) ) ) }
%Y A084389 Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.
%Y A084389 Adjacent sequences: A084386 A084387 A084388 this_sequence A084390 A084391 A084392
%Y A084389 Sequence in context: A077385 A066013 A014521 this_sequence A122022 A049258 A078076
%K A084389 easy,nonn
%O A084389 1,1
%A A084389 Cino Hilliard (hillcino368(AT)gmail.com), Jun 23 2003

%I A122022
%S A122022 0,1,2,1,1,3,13,19,26,2,115,305,591,615,880,5150,14015,23855,8895,83805,
%T A122022 350090,827190,1013985,829725,8881795,28734355,54083980,32511130,
%U A122022 207297335,1011859275,2580294695
%V A122022 0,1,2,1,1,-3,-13,-19,-26,-2,115,305,591,615,-880,-5150,-14015,-23855,-8895,83805,
%W A122022 350090,827190,1013985,-829725,-8881795,-28734355,-54083980,-32511130,207297335,
%X A122022 1011859275,2580294695
%N A122022 a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 1; for n > 3, a(n) = a(n - 1) - (n - 1)*a(n - 4).
%t A122022 a[0] = 0; a[1] = 1; a[2] = 2; a[3] = 1; a[n_] := a[n] = a[n - 1] - (n - 1)*a[n - 4] Table[a[n], {n, 0, 30}]
%Y A122022 Cf. A000898, A121966, A062267.
%Y A122022 Adjacent sequences: A122019 A122020 A122021 this_sequence A122023 A122024 A122025
%Y A122022 Sequence in context: A066013 A014521 A084389 this_sequence A049258 A078076 A010247
%K A122022 sign
%O A122022 1,3
%A A122022 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 12 2006
%E A122022 Edited by njas, Sep 12 2006

%I A049258
%S A049258 2,1,1,3,14,3,3,4,10,1,8,4,6,5,1,3,2,1,0,19,19,6,19,3,19,2,1,3,11,19,1,
%T A049258 11,7,19,19,19,6,19,13,4,7,3,19,19,19,19,13,16,3,1,1,7,19,19,19,19,19,
%U A049258 2,1,4,3,2,13,3,11,10,1,8,7,5,1,4,3,2,1,0,19,19,1,19,10,3,8,19,15,5,3
%N A049258 Smallest nonnegative value taken on by 19x^2 - ny^2 for an infinite number of integer pairs (x, y).
%Y A049258 Adjacent sequences: A049255 A049256 A049257 this_sequence A049259 A049260 A049261
%Y A049258 Sequence in context: A014521 A084389 A122022 this_sequence A078076 A010247 A087605
%K A049258 nonn
%O A049258 1,1
%A A049258 David W. Wilson (davidwwilson(AT)comcast.net)

%I A078076
%S A078076 0,2,1,1,3,54,1,1,5,2,3,2,2,2,2,2,1,3,32,2,1,2,17,1,1,1,1,14,4,1,19,
%T A078076 4,2,3,40,1,1,2,1,19,2,3,5324,2,1,3,2,1,5,3,1,1,17,1,1,2,8,1,2,2,3,
%U A078076 2,2,1,2,15,1,7,1,5,2,1,4,16,272,1,8,3,1,2,3,3,39,2,2,1,4,1,1,1,2,3
%N A078076 Continued fraction for constant defined in A065481.
%Y A078076 Adjacent sequences: A078073 A078074 A078075 this_sequence A078077 A078078 A078079
%Y A078076 Sequence in context: A084389 A122022 A049258 this_sequence A010247 A087605 A106246
%K A078076 nonn,cofr
%O A078076 1,2
%A A078076 Benoit Cloitre, Dec 02 2002

%I A010247
%S A010247 2,1,1,3,138,1,1,3,2,3,1,1,207,1,2,2,1,1,1,1,2,4,9,1,2,
%T A010247 4,1,1,3,4,277,2,5,3,3,3,1,1,1,1,13,2,15,20,2,1,1,1,1,1,
%U A010247 2,1,2,18,2,4,1,22,20,51,23,2,1,3,2,204,1,2,3,1,4,1,3,76
%N A010247 Continued fraction for cube root of 17.
%H A010247 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%Y A010247 Adjacent sequences: A010244 A010245 A010246 this_sequence A010248 A010249 A010250
%Y A010247 Sequence in context: A122022 A049258 A078076 this_sequence A087605 A106246 A136674
%K A010247 nonn,cofr
%O A010247 0,1
%A A010247 njas

%I A087605
%S A087605 1,2,1,1,3,100005,1,2,4,6,8,100010,19,2,215,9,60,100041,4,66,5,1,41,
%T A087605 100061,4,15,2,1,195,100055,61,1061,143,12,72,100127,19,60,1,6,125,0,45,
%U A087605 1305,3,39,27,100269,72,95,136,1123,50,100193,52,1056,176,1536,66
%N A087605 Smallest k such that n times concatenation of k with itself followed by a 7 is a prime, or 0 if no such number exists.
%C A087605 a(42n)=0, but all other terms are probably nonzero. For n a multiple of 42, (10^(l*n)-1)/(10^l-1)*10+7 is divisible by 7 for any l. - Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 11 2005
%F A087605 Minimal k such that k*(10^(l*n)-1)/(10^l-1)*10+7 is prime, where l is the length of k; and 0 if no such prime exists. - Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 11 2005
%e A087605 a(5) = 3 as 333337 is a prime but 111117 and 222227 are not.
%o A087605 (PARI) { a(n) = if(n%42==0,return(0)); for(l=1,10^6, if(valuation(10^(l*n)-1,7)==valuation(10^l-1,7), for(k=10^(l-1),10^l-1, if(isprime(k*(10^(l*n)-1)/(10^l-1)*10+7), return(k) ) ) ) ) } (Alekseyev)
%Y A087605 Cf. A086920, A087604, A087606, A087607, A087608, A087609, A087610.
%Y A087605 Adjacent sequences: A087602 A087603 A087604 this_sequence A087606 A087607 A087608
%Y A087605 Sequence in context: A049258 A078076 A010247 this_sequence A106246 A136674 A064645
%K A087605 base,nonn
%O A087605 1,2
%A A087605 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 18 2003
%E A087605 Corrected and extended by Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 11 2005

%I A106246
%S A106246 1,2,1,1,4,1,0,3,6,1,0,0,6,8,1,0,0,0,10,10,1,0,0,0,0,15,12,1,0,0,0,0,0,
%T A106246 21,14,1,0,0,0,0,0,0,28,16,1,0,0,0,0,0,0,0,36,18,1,0,0,0,0,0,0,0,0,45,
%U A106246 20,1,0,0,0,0,0,0,0,0,0,55,22,1,0,0,0,0,0,0,0,0,0,0,66,24,1,0,0,0,0,0,0
%N A106246 Number triangle T(n,k)=C(n,k)C(2,n-k).
%C A106246 Row sums are C(n+2,2)=A000217(n+1). Diagonal sums are A106247. Subdiagonal is 2(n+1)=A005843(n+1). Second subdiagonal is A000217(n+1).
%e A106246 Triangle begins
%e A106246 1;
%e A106246 2,1;
%e A106246 1,4,1;
%e A106246 0,3,6,1;
%e A106246 0,0,6,8,1;
%e A106246 0,0,0,10,10,1;
%Y A106246 Adjacent sequences: A106243 A106244 A106245 this_sequence A106247 A106248 A106249
%Y A106246 Sequence in context: A078076 A010247 A087605 this_sequence A136674 A064645 A008307
%K A106246 easy,nonn,tabl
%O A106246 0,2
%A A106246 Paul Barry (pbarry(AT)wit.ie), Apr 26 2005

%I A136674
%S A136674 1,2,1,1,4,1,0,8,6,1,1,12,19,8,1,2,15,44,34,10,1,3,16,84,104,53,12,1,4,
%T A136674 14,140,258,200,76,14,1,5,8,210,552,605,340,103,16,1,6,3,288,1056,1562,
%U A136674 1209,532,134,18,1,7,20,363,1848,3575,3640,2170,784,169,20,1
%V A136674 1,2,-1,1,-4,1,0,-8,6,-1,-1,-12,19,-8,1,-2,-15,44,-34,10,-1,-3,-16,84,-104,53,-12,1,-4,
%W A136674 -14,140,-258,200,-76,14,-1,-5,-8,210,-552,605,-340,103,-16,1,-6,3,288,-1056,1562,
%X A136674 -1209,532,-134,18,-1,-7,20,363,-1848,3575,-3640,2170,-784,169,-20,1
%N A136674 Triangular sequence made from matrices of the type( Cartan G_n types): M(3)= {{2, -1, 0}, {-1, 2, -1}, {0, -3, 2}} with polynomial recursion: p(x, n) = (2 - x)*p(x, n - 1) - p(x, n - 2).
%C A136674 Row sums:
%C A136674 {1, 1, -2, -3, -1, 2, 3, 1, -2, -3, -1}
%C A136674 The 3 X 3 matrix with determinant zero is forbidden in the Cartan definition in the same way as E_9 is forbidden.
%F A136674 T(n, m, d) = If[ n == m, 2, If[n == d && m == d - 1, -3, If[(n == m - 1 || n == m + 1), -1, 0]]] or p(x, n) = (2 - x)*p(x, n - 1) - p(x, n - 2)
%e A136674 {1},
%e A136674 {2, -1},
%e A136674 {1, -4, 1},
%e A136674 {0, -8, 6, -1},
%e A136674 {-1, -12, 19, -8, 1},
%e A136674 {-2, -15, 44, -34, 10, -1},
%e A136674 {-3, -16, 84, -104, 53, -12, 1},
%e A136674 {-4, -14, 140, -258, 200, -76, 14, -1},
%e A136674 {-5, -8, 210, -552,605, -340, 103, -16, 1},
%e A136674 {-6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1},
%e A136674 {-7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1}
%t A136674 (* tridiagonal matrix code*) T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -3, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a0 = Table[M[d], {d, 1, 10}]; Table[Det[M[d]], {d, 1, 10}]; g = Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] MatrixForm[a]; (* polynomial recursion: three initial terms necessary*) Clear[p] p[x, 0] = 1; p[x, 1] = (2 - x); p[x, 2] = 1 - 4 x + x^2; p[x_, n_] := p[x, n] = (2 - x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
%Y A136674 Adjacent sequences: A136671 A136672 A136673 this_sequence A136675 A136676 A136677
%Y A136674 Sequence in context: A010247 A087605 A106246 this_sequence A064645 A008307 A099238
%K A136674 nonn,uned,tabl
%O A136674 1,2
%A A136674 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 05 2008

%I A064645
%S A064645 1,1,1,2,1,1,4,1,1,1,9,2,1,1,1,21,4,1,1,1,1,51,8,2,1,1,1,1,127,17,4,1,1,1,1,1,323,37,8,2,
%T A064645 1,1,1,1,1,835,82,16,4,1,1,1,1,1,1,2188,185,33,8,2,1,1,1,1,1,1,5798,423,69,16,4,1,1,1,1,
%U A064645 1,1,1,15511,978,146,32,8,2,1,1,1,1,1,1,1,41835,2283,312,65,16,4,1,1,1,1,1,1,1,1
%N A064645 Table where the entry (n,k) (n >= 0,k >= 0) gives number of Motzkin paths of the length n with the minimum peak width of k.
%e A064645 E.g. we have the following nine Motzkin paths of length 4, of which 4 last have all peaks at least of width 1, and two last with any peak at least 2 dashes wide, so M(4,0) = 9, M(4,1) = 4 and M(4,2) = 2.
%e A064645 ./\............................_......_....__
%e A064645 /..\../\/\..__/\.._/\_../\__../.\_.._/.\../..\..____
%p A064645 [seq(A064645(j),j=0..104)]; A064645 := (n) -> Mpw((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)));
%p A064645 C := (n,k) -> `if`((n <= 0),0,binomial(n,k));
%p A064645 Mpw := proc(n,m) local i,k; 1+add(add(A001263(i,k)*C(n-(m*k),2*i),k=1..i),i=0..floor(n/2)); end;
%Y A064645 First row (k=0): Motzkin numbers (A001006), second row (k=1): A004148 (with RNA connection), third row (k=2): A004149, fourth row (k=3): A023421, fifth row (k=4): A023422, sixth row (k=5): A023423. Uses the table A001263(n, k) which gives the Dyck paths (Catalan Mountain Ranges) with exactly k peaks. Maple procedure trinv given at A054425.
%Y A064645 Adjacent sequences: A064642 A064643 A064644 this_sequence A064646 A064647 A064648
%Y A064645 Sequence in context: A087605 A106246 A136674 this_sequence A008307 A099238 A061462
%K A064645 nonn,tabl
%O A064645 0,4
%A A064645 Antti Karttunen Oct 03 2001

%I A008307
%S A008307 1,1,1,1,2,1,1,4,1,1,1,10,3,2,1,1,26,9,4,1,1,1,76,21,16,1,2,1,1,232,81,
%T A008307 56,1,6,1,1,1,764,351,256,25,18,1,2,1,1,2620,1233,1072,145,66,1,4,1,1,
%U A008307 1,9496,5769,6224,505,396,1,16,3,2,1,1,35696,31041,33616,1345,2052,1
%N A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.
%C A008307 Solutions to x^k = 1 in Symm_n.
%D A008307 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
%F A008307 T(n+1, k) = Sum_{d|k} (n)_{d-1}*T(n-d+1, k), where (n)_i = n*(n-1)*(n-2)*...*(n-i+1).
%F A008307 Sum_{n >= 0} T(n, k)*t^n/n! = exp( sum_{d|k} t^d/d ).
%e A008307 Array begins
%e A008307 1 1 1 1 1 1 ...
%e A008307 1 2 1 2 1 2 ...
%e A008307 1 4 9 4 1 6 ...
%e A008307 1 10 9 16 1 ...
%e A008307 1 26 21 56 25 ...
%Y A008307 Rows give A056595, (more sequences needed!), columns give A000085, A001470, A001472, A052501, A053496-A053505.
%Y A008307 Adjacent sequences: A008304 A008305 A008306 this_sequence A008308 A008309 A008310
%Y A008307 Sequence in context: A106246 A136674 A064645 this_sequence A099238 A061462 A122578
%K A008307 nonn,tabl,easy,nice
%O A008307 1,5
%A A008307 njas
%E A008307 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 13 2001

%I A099238
%S A099238 1,1,2,1,1,4,1,1,2,8,1,1,1,3,16,1,1,1,2,5,32,1,1,1,1,3,8,64,1,1,1,1,2,4,
%T A099238 13,128,1,1,1,1,1,3,6,21,256,1,1,1,1,1,2,4,9,34,512,1,1,1,1,1,1,3,5,13,
%U A099238 55,1024,1,1,1,1,1,1,2,4,7,19,89,2048
%N A099238 Square array read by anti-diagonals with rows generated by 1/(1-x-x^(k+1)).
%C A099238 Sections of rows are given by array A099233. Sums of anti-diagonals yield A097939.
%F A099238 Square array T(n, k)=sum{j=0..floor(n/(k+1)), binomial(n-k*j, j)}, n, k>=0.
%e A099238 Rows begin
%e A099238 1,2,4,8,16,32,.. (A000079)
%e A099238 1,1,2,3,5,8,13,... (A000045)
%e A099238 1,1,1,2,3,4,6,9,... (A000930)
%e A099238 1,1,1,1,2,3,4,5,7,.. (A003269)
%e A099238 1,1,1,1,1,2,3,4,5,.. (A003520)
%Y A099238 Adjacent sequences: A099235 A099236 A099237 this_sequence A099239 A099240 A099241
%Y A099238 Sequence in context: A136674 A064645 A008307 this_sequence A061462 A122578 A005131
%K A099238 easy,nonn,tabl
%O A099238 0,3
%A A099238 Paul Barry (pbarry(AT)wit.ie), Oct 08 2004

%I A061462
%S A061462 1,1,2,1,1,4,1,1,4,1,1,2,1,1,2,1,1,4,1,1,4,1,1,2,1,1,2,1,1,4,1,1,4,1,1,2,
%T A061462 1,1,2,1,1,4,1,1,4,1,1,2,1,1,2,1,1,4,1,1,4,1,1,2,1,1,2,1,1,4,1,1,4,1,1,2,
%U A061462 1,1,2,1,1,4,1,1,4,1,1,2,1,1,2,1,1,4,1,1,4,1,1,2,1,1,2,1,1,4,1,1,4,1,1,2
%N A061462 The exact power of 2 that divides the n-th Bell number (A000110). Has period 12.
%C A061462 { Bell(n) mod 8 } is periodic with period 24, the period being (1 1 2 5 7 4 3 5 4 3 7 2 5 5 2 1 3 4 7 1 4 7 3 2). Hence the highest power of 2 dividing a Bell number is 4. - David W. Wilson (davidwwilson(AT)comcast.net), Jun 29, 2001
%D A061462 W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979) 1-16.
%Y A061462 Cf. A000110.
%Y A061462 Adjacent sequences: A061459 A061460 A061461 this_sequence A061463 A061464 A061465
%Y A061462 Sequence in context: A064645 A008307 A099238 this_sequence A122578 A005131 A105477
%K A061462 nonn
%O A061462 0,3
%A A061462 Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 10 2001

%I A122578
%S A122578 1,2,1,1,4,1,1,4,29,1,20,29,299,20,559,299,3926,559,13429,3926,61165,13429,
%T A122578 343174,61165,1063621,343174,9642971,1063621,19074796,9642971,298720955,19074796,
%U A122578 292597721,298720955,10150389236,292597721,90530999,10150389236,375654932731
%V A122578 1,2,1,-1,-4,-1,-1,4,29,1,-20,-29,-299,20,559,299,3926,-559,-13429,-3926,-61165,13429,
%W A122578 343174,61165,1063621,-343174,-9642971,-1063621,-19074796,9642971,298720955,19074796,
%X A122578 292597721,-298720955,-10150389236,-292597721,-90530999,10150389236,375654932731
%N A122578 A switch between two types of n level recursion: Even:a(n)=(n - 1)*a(n - 1) - a(n - 2) Odd:a(n)=a(n - 1) - (n - 2)*a(n - 2).
%F A122578 Even:a(n)=(n - 1)*a(n - 1) - a(n - 2) Odd:a(n)=a(n - 1) - (n - 2)*a(n - 2)
%t A122578 a[0] = 1; a[1] = 2; a[n_] := a[n] = If[Mod[n, 2] == 0, (n - 1)*a[n - 1] - a[n - 2], a[n - 1] - (n -2)*a[n - 2]] Table[a[n], {n, 0, 50}]
%Y A122578 Adjacent sequences: A122575 A122576 A122577 this_sequence A122579 A122580 A122581
%Y A122578 Sequence in context: A008307 A099238 A061462 this_sequence A005131 A105477 A127709
%K A122578 sign,uned
%O A122578 1,2
%A A122578 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 17 2006

%I A005131
%S A005131 1,0,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,1,16,1,1,18,
%T A005131 1,1,20,1,1,22,1,1,24,1,1,26,1,1,28,1,1,30,1,1,32,1,1,34,1,1,36,1,1,
%U A005131 38,1,1,40,1,1,42
%N A005131 A generalized continued fraction for Euler's number e.
%C A005131 Only a(1) = 0 prevents this from being a simple continued fraction. The motivation for this alternate representation is that the simple pattern {1, 2*n, 1} (from n=0) may be more mathematically appealing than the pattern in the corresponding simple continued fraction (at A003417) - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
%D A005131 H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.
%D A005131 Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".
%D A005131 T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
%H A005131 Alexander R. Povolotsky (pevnev(AT)juno.com), Feb 23 2008, <a href="http://www.research.att.com/~njas/sequences/b005131.txt">Table of n, a(n) for n = 1..60</a>
%H A005131 A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a117.pdf">Continued fraction expansions of values of the exponential function...</a>
%H A005131 A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a151.pdf">Number theory and Kustaa Inkeri</a>
%F A005131 If Mod[n,3]==1, a(n) = 2*(k-1)/3, else a(n) = 1. - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
%t A005131 Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
%Y A005131 Cf. A003417, A100261.
%Y A005131 Adjacent sequences: A005128 A005129 A005130 this_sequence A005132 A005133 A005134
%Y A005131 Sequence in context: A099238 A061462 A122578 this_sequence A105477 A127709 A131350
%K A005131 nonn,cofr
%O A005131 0,5
%A A005131 Russ Cox (rsc(AT)swtch.com)

%I A105477
%S A105477 1,2,1,1,4,1,1,6,6,1,1,6,15,8,1,1,7,23,28,10,1,1,8,30,60,45,12,1,1,9,39,
%T A105477 98,125,66,14,1,1,10,49,144,255,226,91,16,1,1,11,60,202,437,561,371,120,
%U A105477 18,1,1,12,72,272,685,1128,1092,568,153,20,1,1,13,85,355,1015,1995,2555
%N A105477 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2.
%F A105477 G.f.=tz(1+z-z^2)/(1-z-tz-tz^2+tz^3).
%F A105477 T(n,k)=Sum(binomial(k,j)*binomial(n-2j-1, k-j-1), j=0..n-k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 06 2006
%e A105477 T(4,2)=6 because we have (1,3),(3,1),(2,2),(2,2'),(2',2) and (2',2').
%e A105477 Triangle begins:
%e A105477 1;
%e A105477 2,1;
%e A105477 1,4,1;
%e A105477 1,6,6,1;
%e A105477 1,6,15,8,1;
%p A105477 G:=t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3): Gser:=simplify(series(G,z=0,15)): for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 13 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form
%Y A105477 Row sums yield A077998.
%Y A105477 Adjacent sequences: A105474 A105475 A105476 this_sequence A105478 A105479 A105480
%Y A105477 Sequence in context: A061462 A122578 A005131 this_sequence A127709 A131350 A131087
%K A105477 nonn,tabl
%O A105477 1,2
%A A105477 Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005

%I A127709
%S A127709 1,1,2,1,1,4,1,1,7,9,1,4,28,15,1,2,47,91,26,1,27,268,257,40,10,312,1318,
%T A127709 643,1,137,2807,5347,1,35,2204,19516,5,771,26312,2,186,14758,39,4362,11,
%U A127709 1013,1,214,1,43,5,2
%N A127709 Table T(d,n) = the number of isomorphism classes of smooth Fano d-polytopes with n vertices, read by rows.
%C A127709 Abstract: "We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes for d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes." The column sums make a good sequence, too.
%H A127709 Mikkel Obro, <a href="http://arxiv.org/pdf/0704.0049">An algorithm for the classification of smooth Fano polytopes</a>, Apr 02 2007, p. 15.
%e A127709 Table begins:
%e A127709 n..|d=1|d=2|d=3|d=4|d=5|d=6|d=7
%e A127709 1..|...........................
%e A127709 2..|.1.|.......................
%e A127709 3..|...|.1.|...................
%e A127709 4..|...|.2.|.1.|...............
%e A127709 5..|...|.1.|.4.|.1.|...........
%e A127709 6..|...|.1.|.7.|.9.|.1.|.......
%e A127709 7..|...|...|.4.|.28|.15|.1.|...
%e A127709 8..|...|...|.2.|.47|.91|.26|.1.
%e A127709 9..|...|...|...|.27|268|257|.40
%Y A127709 Adjacent sequences: A127706 A127707 A127708 this_sequence A127710 A127711 A127712
%Y A127709 Sequence in context: A122578 A005131 A105477 this_sequence A131350 A131087 A105475
%K A127709 nonn,tabl
%O A127709 1,3
%A A127709 Jonathan Vos Post (jvospost2(AT)yahoo.com), Apr 03 2007

%I A131350
%S A131350 1,2,1,1,4,1,2,4,6,1,1,8,9,8,1,2,7,20,16,10,1,1,12,24,40,25,12,1,2,10,
%T A131350 42,60,70,36,14,1,1,16,46,112,125,112,49,16,1,2,13,72,148,252,231,168,
%U A131350 64,18,1
%N A131350 2*A007318 - A049310 as infinite lower triangular matrices.
%C A131350 Row sums = A0099036 starting (1, 3, 6, 13, 27, 56, 115,...).
%e A131350 First few rows of the triangle are:
%e A131350 1;
%e A131350 2, 1;
%e A131350 1, 4, 1;
%e A131350 2, 4, 6, 1;
%e A131350 1, 8, 9, 8, 1;
%e A131350 2, 7, 20, 16, 10, 1;
%e A131350 1, 12, 24, 40, 25, 12, 1;
%e A131350 ...
%Y A131350 Cf. A007318, A049310, A131350.
%Y A131350 Adjacent sequences: A131347 A131348 A131349 this_sequence A131351 A131352 A131353
%Y A131350 Sequence in context: A005131 A105477 A127709 this_sequence A131087 A105475 A136321
%K A131350 nonn,tabl
%O A131350 0,2
%A A131350 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 02 2007

%I A131087
%S A131087 1,2,1,1,4,1,2,5,6,1,1,8,11,8,1,2,9,20,19,10,1,1,12,29,40,29,12,1,2,13,
%T A131087 42,69,70,41,14,1,1,16,55,112,139,112,55,16,1,2,17,72,167,252,251,168,
%U A131087 71,18,1,1,20,89,240,419,504,419,240,89,20,1,2,21,110,329,660,923,924
%N A131087 Triangle read by rows: T(n,k)=2*binom(n,k)-[1+(-1)^(n-k)]/2 (0<=k<=n).
%C A131087 Row sums = A084174: (1, 3, 6, 14, 29,...).
%C A131087 2*A007318 - A128174 as infinite lower triangular matrices. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2007
%F A131087 G.f.=G(t,z)=(1+z-tz-2z^2+2tz^3)/[(1-z^2)(1-tz)(1-z-tz)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2007
%e A131087 First few rows of the triangle are:
%e A131087 1;
%e A131087 2, 1;
%e A131087 1, 4, 1;
%e A131087 2, 5, 6, 1;
%e A131087 1, 8, 11, 8, 1;
%e A131087 2, 9, 20, 19, 10, 1;
%e A131087 1, 12, 29, 40, 29, 12, 1;
%e A131087 ...
%p A131087 T := proc (n, k) options operator, arrow; 2*binomial(n, k)-1/2-(1/2)*(-1)^(n-k) end proc; for n from 0 to 11 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2007
%Y A131087 Cf. A128174, A084174.
%Y A131087 Adjacent sequences: A131084 A131085 A131086 this_sequence A131088 A131089 A131090
%Y A131087 Sequence in context: A105477 A127709 A131350 this_sequence A105475 A136321 A112987
%K A131087 nonn,tabl
%O A131087 0,2
%A A131087 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2007
%E A131087 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2007

%I A105475
%S A105475 1,2,1,1,4,1,2,6,6,1,1,8,15,8,1,2,11,26,28,10,1,1,12,42,64,45,12,1,2,16,
%T A105475 60,122,130,66,14,1,1,16,82,208,295,232,91,16,1,2,21,108,324,582,621,
%U A105475 378,120,18,1,1,20,135,480,1035,1404,1176,576,153,20,1,2,26,170,675
%N A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.
%e A105475 T(4,2)=6 because we have (1,3),(3,1),(2,2),(2,2'),(2',2) and (2',2').
%e A105475 Triangle begins:
%e A105475 1;
%e A105475 2,1;
%e A105475 1,4,1;
%e A105475 2,6,6,1;
%e A105475 1,8,15,8,1;
%p A105475 G:=t*z*(1+2*z)/(1-t*z-z^2-2*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form
%Y A105475 Row sums yield A105476.
%Y A105475 Adjacent sequences: A105472 A105473 A105474 this_sequence A105476 A105477 A105478
%Y A105475 Sequence in context: A127709 A131350 A131087 this_sequence A136321 A112987 A125138
%K A105475 nonn,tabl
%O A105475 1,2
%A A105475 Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005

%I A136321
%S A136321 1,2,1,1,4,1,4,6,6,1,7,4,17,8,1,10,5,32,32,10,1,13,24,42,88,51,12,1,16,
%T A136321 56,28,186,180,74,14,1,19,104,42,312,495,316,101,16,1,22,171,216,396,
%U A136321 1122,1053,504,132,18,1,25,260,561,264,2145,2912,1960,752,167,20,1
%V A136321 1,-2,1,-1,-4,1,4,6,-6,1,-7,-4,17,-8,1,10,-5,-32,32,-10,1,-13,24,42,-88,51,-12,1,16,
%W A136321 -56,-28,186,-180,74,-14,1,-19,104,-42,-312,495,-316,101,-16,1,22,-171,216,396,-1122,
%X A136321 1053,-504,132,-18,1,-25,260,-561,-264,2145,-2912,1960,-752,167,-20,1
%N A136321 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4*x+4-m:m=5;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.
%C A136321 Row sums are:
%C A136321 {1, -1, -4, 5, -1, -4, 5, -1, -4, 5, -1}
%C A136321 This sequence is also related to different p(x,2) start:
%C A136321 1) A_n like sequence A053122 ( sign change)
%C A136321 2) my G_n matrix A136674
%C A136321 3) B_n,C_n A110162
%F A136321 p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) Three start vectors necessary: p(x,0)=1;p(x,1)=2-x; p(x,2)=x^2-4*x-1=CharacteristicPolynomial[{{2, -5}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-5, 2}}, x]
%e A136321 {1},
%e A136321 {-2, 1},
%e A136321 {-1, -4, 1},
%e A136321 {4, 6, -6, 1},
%e A136321 {-7, -4, 17, -8, 1},
%e A136321 {10, -5, -32, 32, -10, 1},
%e A136321 {-13, 24, 42, -88,51, -12, 1},
%e A136321 {16, -56, -28,186, -180, 74, -14, 1},
%e A136321 {-19, 104, -42, -312, 495, -316, 101, -16, 1},
%e A136321 {22, -171, 216, 396, -1122, 1053, -504, 132, -18, 1},
%e A136321 {-25, 260, -561, -264,2145, -2912, 1960, -752, 167, -20, 1}
%t A136321 Clear[p, a] p[x, 0] = 1; p[x, 1] = -2 + x; p[x, 2] = x^2 - 4*x - 1; p[x_, n_] := p[x, n] = (-2 + x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}] Flatten[a]
%Y A136321 Cf. A053122, A136674, A110162.
%Y A136321 Adjacent sequences: A136318 A136319 A136320 this_sequence A136322 A136323 A136324
%Y A136321 Sequence in context: A131350 A131087 A105475 this_sequence A112987 A125138 A021477
%K A136321 nonn,tabl,uned,new
%O A136321 1,2
%A A136321 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2008

%I A112987
%S A112987 2,1,1,4,1,4,16,4,1,256,16,4,16,4,16,256,1,4,1024,4,65536,256,16,4,
%T A112987 65536,128,16,67108864,65536,4,16,4,1,256,16,262144,268435456,4,16,256,
%U A112987 65536,4,4194304,4,65536,131072,16,4,65536,1073741824,16777216,256
%N A112987 2^(2^n mod n).
%F A112987 a(n)=2^A112986(n)
%Y A112987 Adjacent sequences: A112984 A112985 A112986 this_sequence A112988 A112989 A112990
%Y A112987 Sequence in context: A131087 A105475 A136321 this_sequence A125138 A021477 A124939
%K A112987 easy,nonn
%O A112987 0,1
%A A112987 Paul Barry (pbarry(AT)wit.ie), Oct 08 2005

%I A125138
%S A125138 1,2,1,1,4,1,5,1,12,19,1,24,32,19,1,20,1,1,20,1,7,57,1,1,6,83,1,15,33,1,
%T A125138 38,9,1,23,70,71,57,17,1,26,1,1,28,1,1,56,67,1,1,73,1,75,1,114,177,76,1,
%U A125138 137,1,76,29,172,132,87,265,1,52,142,9,76,1,311,1,209,37,149,115,227,1
%V A125138 -1,2,-1,-1,4,-1,5,-1,12,19,-1,24,32,19,-1,20,-1,-1,20,-1,7,57,-1,-1,6,83,-1,15,33,-1,
%W A125138 38,9,-1,23,70,71,57,17,-1,26,-1,-1,28,-1,-1,56,67,-1,-1,73,-1,75,-1,114,177,76,-1,137,
%X A125138 -1,76,29,172,132,87,265,-1,52,142,9,76,-1,311,-1,209,37,149,115,227,-1,370,-1,333,-1
%N A125138 Smarandache-Wagstaff function: a(n) = smallest m such that prime(n) divides Sum_{i=1..m} i!, or -1 if no such m exists.
%C A125138 One need only check values of m < prime(n).
%C A125138 This takes values -1 at A056985 and values given in A056984 at the primes listed in A056983.
%D A125138 F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House, 2000.
%H A125138 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Definitions-book.pdf">Definitions, Solved and Unsolved Problems, Conjectures, ... </a>
%H A125138 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Smarandache-WagstaffFunction.html">Link to a section of The World of Mathematics.</a>
%p A125138 A007489 := proc(n) add(factorial(k),k=1..n) ; end: A125138 := proc(n) local p,m ; p := ithprime(n) ; for m from 1 to p do if A007489(m) mod p = 0 then RETURN(m) ; fi ; od ; RETURN(-1) ; end: for n from 1 to 120 do printf("%d,",A125138(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 14 2007
%Y A125138 Cf. A007489, A056983, A056984, A056985.
%Y A125138 Adjacent sequences: A125135 A125136 A125137 this_sequence A125139 A125140 A125141
%Y A125138 Sequence in context: A105475 A136321 A112987 this_sequence A021477 A124939 A099020
%K A125138 sign
%O A125138 1,2
%A A125138 njas, Jan 21 2007
%E A125138 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 14 2007

%I A021477
%S A021477 0,0,2,1,1,4,1,6,4,9,0,4,8,6,2,5,7,9,2,8,1,1,8,3,9,3,2,3,4,6,7,2,3,
%T A021477 0,4,4,3,9,7,4,6,3,0,0,2,1,1,4,1,6,4,9,0,4,8,6,2,5,7,9,2,8,1,1,8,3,
%U A021477 9,3,2,3,4,6,7,2,3,0,4,4,3,9,7,4,6,3,0,0,2,1,1,4,1,6,4,9,0,4,8,6,2
%N A021477 Decimal expansion of 1/473.
%Y A021477 Adjacent sequences: A021474 A021475 A021476 this_sequence A021478 A021479 A021480
%Y A021477 Sequence in context: A136321 A112987 A125138 this_sequence A124939 A099020 A089688
%K A021477 nonn,cons
%O A021477 0,3
%A A021477 njas

%I A124939
%S A124939 1,1,1,2,1,1,4,1,6,5,1,1,10,1,12,7,1,16,3,8,1,1,18,1,22,9,1,28,13,24,1,
%T A124939 30,11,20,17,1,1,36,1,40,19,1,42,25,34,1,46,15,14,23,1,52,21,26,27,32,1,
%U A124939 1,58,1,60,29,1,66,31,48,1,70,33,38,35,1,72,37,64,39,44,1,78,49,54,43
%N A124939 Prime tetrahedron, read by rows.
%C A124939 Each triangular layer of the unique tetrahedron begins with 1, never uses any value other than 1 which has occurred already on this or earlier levels, always uses the least available integer such that the sum of each two consecutive entries is a prime. The number of values of the n-th level is the n-th triangular number A000217(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. The number of values through the n-th level is the n-th tetrahedral number A000292(n) = C(n+2,3) = n(n+1)(n+2)/6.
%D A124939 R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.
%D A124939 Kenney, M. J. "Student Math Notes." NCTM News Bulletin. Nov. 1986.
%H A124939 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeTriangle.html">Prime Triangle</a>.
%F A124939 a(n) flattens the 3-D table so that level 1 (the apex, with only the value 1) occures first, then level 2 (with values 1, 1, 2), then level 3 ... and for each level, reads that triangle by rows.
%e A124939 Tetrahedron begins
%e A124939 =================
%e A124939 1
%e A124939 =================
%e A124939 1
%e A124939 1..2
%e A124939 =================
%e A124939 1
%e A124939 1..4
%e A124939 1..6..5
%e A124939 =================
%e A124939 1
%e A124939 1.10
%e A124939 1.12..7
%e A124939 1.16..3..8
%e A124939 =================
%e A124939 1
%e A124939 1.18
%e A124939 1.22..9
%e A124939 1.28.13.24
%e A124939 1.30.11.20.17
%e A124939 =================
%p A124939 srch := proc(a) local res ; res := 2 ; while true do if isprime(res+op(-1,a)) and not ( res in a ) then RETURN(res) ; fi ; res := res+1 ; od ; end: a := [] ; for lvl from 1 to 10 do for row from 1 to lvl do for col from 1 to row do if col = 1 then anxt := 1 ; else anxt := srch(a) ; fi ; printf("%d,",anxt) ; a := [op(a), anxt] ; od ; od ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 13 2007
%Y A124939 Cf. A000040, A000217, A000292, A036440 Number of ways of arranging row n of the Prime Pyramid, A051239, A051237 Lexicographically earliest Prime Pyramid, read by rows.
%Y A124939 Adjacent sequences: A124936 A124937 A124938 this_sequence A124940 A124941 A124942
%Y A124939 Sequence in context: A112987 A125138 A021477 this_sequence A099020 A089688 A092479
%K A124939 easy,nonn
%O A124939 1,4
%A A124939 Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 13 2006
%E A124939 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 13 2007

%I A099020
%S A099020 1,1,0,2,1,1,4,2,1,0,10,6,4,3,3,26,16,10,6,3,0,76,50,34,24,18,15,15,232,
%T A099020 156,106,72,48,30,15,0,764,532,376,270,198,150,120,105,105,2620,1856,
%U A099020 1324,948,678,480,330,210,105,0,9496,6876,5020,3696,2748,2070
%N A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.
%C A099020 In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.
%H A099020 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%F A099020 Recurrence: T(0, 2n) = (2n-1)!!, T(0, 2n+1) = 0, T(k, n) = T(k-1, n) + T(k-1, n+1).
%e A099020 1,0,1,0,3,0,15,
%e A099020 1,1,1,3,3,15,15,
%e A099020 2,2,4,6,18,30,120,
%e A099020 4,6,10,24,48,150,330,
%e A099020 10,16,34,72,198,480,1590,
%Y A099020 First column is A000085, main diagonal is in A099021.
%Y A099020 Adjacent sequences: A099017 A099018 A099019 this_sequence A099021 A099022 A099023
%Y A099020 Sequence in context: A125138 A021477 A124939 this_sequence A089688 A092479 A124022
%K A099020 nonn,tabl
%O A099020 0,4
%A A099020 Ralf Stephan, Sep 23 2004

%I A089688
%S A089688 1,1,1,1,2,1,1,4,2,1,1,6,3,2,1,1,10,5,3,2,1,1,14,7,4,3,2,1,1,20,9,6,4,3,
%T A089688 2,1,1,26,12,8,5,4,3,2,1,1,36,15,10,7,5,4,3,2,1,1,46,18,12,9,6,5,4,3,2,
%U A089688 1,1,60,23,15,11,8,6,5,4,3,2,1
%N A089688 Table T(n,k), n>=0 and k>=1, read by antidiagonals; the k-th row is defined by : partitions of kn into powers of k (with T(0,k) = 1).
%e A089688 Row k = 1 : 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (see A000012).
%e A089688 Row k = 2 : 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, ...(see A000123).
%e A089688 Row k = 3 : 1, 2, 3, 5, 7, 9, 12, 15, 18, 23, 28, 33, ...(see A005704).
%e A089688 Row k = 4 : 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, ...(see A005705).
%e A089688 Row k = 5 : 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, ...(see A005706).
%Y A089688 Adjacent sequences: A089685 A089686 A089687 this_sequence A089689 A089690 A089691
%Y A089688 Sequence in context: A021477 A124939 A099020 this_sequence A092479 A124022 A098063
%K A089688 easy,nonn,tabl
%O A089688 0,5
%A A089688 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 05 2004

%I A092479
%S A092479 1,1,1,1,2,1,1,4,2,1,1,6,6,2,1,1,11,10,7,2,1,1,18,22,13,7,2,1,1,31,42,
%T A092479 30,14,7,2,1,1,54,82,60,34,15,7,2,1,1,97,157,125,71,36,15,7,2,1,1,172,
%U A092479 304,256,152,77,37,15,7,2,1,1,309,589,513,325,168,81,37,15,7,2,1,1,564
%N A092479 T(n,k) = number of numbers <= 2^n having exactly k prime factors (with repetitions), 0<=k<=n, triangle read by rows.
%C A092479 T(n,0)=1; T(n,1)=A007053(n,1) for n>0; T(n,n)=1;
%C A092479 Sum(T(n,k): 0<=k<=n) = 2^n.
%H A092479 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html#primepop">Index entries for sequences related to numbers of primes in various ranges</a>
%Y A092479 Cf. A001222.
%Y A092479 Adjacent sequences: A092476 A092477 A092478 this_sequence A092480 A092481 A092482
%Y A092479 Sequence in context: A124939 A099020 A089688 this_sequence A124022 A098063 A106396
%K A092479 nonn,tabl
%O A092479 0,5
%A A092479 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 27 2004

%I A124022
%S A124022 1,1,1,1,2,1,1,4,2,1,1,6,7,2,1,1,9,12,10,2,1,1,12,26,18,13,2,1,1,16,40,
%T A124022 52,24,16,2,1,1,20,70,86,87,30,19,2,1,1,25,100,190,150,131,36,22,2,1,1,30,
%U A124022 155,294,403,232,184,42,25,2,1,1,36,210,553,656,736,332,246,48,28,2,1,1,42
%V A124022 1,1,-1,-1,2,1,-1,4,2,-1,1,-6,-7,2,1,1,-9,-12,10,2,-1,-1,12,26,-18,-13,2,1,-1,16,40,
%W A124022 -52,-24,16,2,-1,1,-20,-70,86,87,-30,-19,2,1,1,-25,-100,190,150,-131,-36,22,2,-1,-1,30,
%X A124022 155,-294,-403,232,184,-42,-25,2,1,-1,36,210,-553,-656,736,332,-246,-48,28,2,-1,1,-42
%N A124022 Triangular sequence from the characteristic polynomials of the SL(n,Z)/ determiants {1,-1} type triantidiagonal 2 center with one upper, -1 side antidiagonal above and below: M(3)={{0, -1, 1}, {-1, 2, -1}, {2, -1, 0}}.
%C A124022 Matrices: {{1}}, {{-1, 1}, {2, -1}}, {{0, -1, 1}, {-1, 2, -1}, {2, -1, 0}}, {{0, 0, -1, 1}, {0, -1, 2, -1}, {-1, 2, -1, 0}, {2, -1, 0, 0}}, {{0, 0, 0, -1, 1}, {0, 0, -1, 2, -1}, {0, -1, 2, -1, 0}, {-1, 2, -1, 0, 0}, {2, -1, 0, 0,0}}
%F A124022 k=2; m(n,m,d)= = Table[If[n +m - 1 == d && n > 1, k, If[n + m == d, -1, If[n + m - 2 == d, -1, If[n == 1 && m == d, k - 1, 0]]]], {n, 1, d}, {m, 1, d}];
%e A124022 Triangular sequence:
%e A124022 {1},
%e A124022 {1, -1},
%e A124022 {-1, 2, 1},
%e A124022 {-1, 4, 2, -1},
%e A124022 {1, -6, -7, 2, 1},
%e A124022 {1, -9, -12, 10, 2, -1},
%e A124022 {-1, 12, 26, -18, -13, 2, 1},
%e A124022 {-1, 16, 40, -52, -24,16, 2, -1},
%e A124022 {1, -20, -70, 86, 87, -30, -19, 2, 1}
%t A124022 k = 2; An[d_] := Table[If[n + m - 1 == d && n > 1, k, If[n + m == d, -1, If[n + m - 2 == d, -1, If[n == 1 &&m == d, k - 1, 0]]]], {n, 1, d}, {m, 1, d}]; Join[An[1], Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
%Y A124022 Adjacent sequences: A124019 A124020 A124021 this_sequence A124023 A124024 A124025
%Y A124022 Sequence in context: A099020 A089688 A092479 this_sequence A098063 A106396 A048004
%K A124022 sign,tabl,uned,probation
%O A124022 1,5
%A A124022 Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 31 2006

%I A098063
%S A098063 1,1,1,1,1,1,2,1,1,4,2,1,1,6,7,2,1,1,9,13,11,2,1,1,12,28,22,16,2,1,1,16,
%T A098063 46,64,33,22,2,1,1,20,80,118,126,46,29,2,1,1,25,120,258,248,225,61,37,2,
%U A098063 1,1,30,185,438,668,460,374,78,46,2,1,1,36,260,813,1231,1506,782,588,97
%N A098063 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and having k level steps at height >0 (can be easily expressed using RNA secondary structure terminology).
%C A098063 Row sums yield the RNA secondary structure numbers (A004148).
%D A098063 I. L. Hofacker, P. Schuster, and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
%D A098063 P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
%D A098063 M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%H A098063 M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08viennot.html">Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire</a>, Sem. Loth. Comb. B08l (1984) 79-86.
%F A098063 G.f.=G=G(t, z) satisfies z(t-tz+tz^2-1+2z-z^2)G^2-(1-2z+z^2+tz)G+1=0.
%e A098063 Triangle starts:
%e A098063 1;
%e A098063 1;
%e A098063 1;
%e A098063 1,1;
%e A098063 1,2,1;
%e A098063 1,4,2,1;
%e A098063 1,6,7,2,1;
%e A098063 1,9,13,11,2,1;
%e A098063 Row n (n>=2) has n-1 terms.
%e A098063 T(5,2)=2 because among the eight peakless Motzkin paths of length 5 only HU(HH)D and U(HH)DH have two H's at positive height (shown between parentheses); here U=(1,1), H=(1,0), D=(1,-1).
%Y A098063 Cf. A004148.
%Y A098063 Adjacent sequences: A098060 A098061 A098062 this_sequence A098064 A098065 A098066
%Y A098063 Sequence in context: A089688 A092479 A124022 this_sequence A106396 A048004 A114394
%K A098063 nonn,tabf
%O A098063 0,7
%A A098063 Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 12 2004

%I A106396
%S A106396 1,1,1,1,2,1,1,4,2,1,1,7,5,2,1,1,11,13,5,2,1,1,16,31,14,5,2,1,1,22,66,
%T A106396 41,14,5,2,1
%N A106396 Triangle read by rows, generated from the Narayana triangle as a matrix.
%C A106396 The n-th column starting from the top has the first n terms in the Catalan sequence: (1, 2, 5, 14...)
%C A106396 First few rows of the triangle are:
%C A106396 1;
%C A106396 1, 1;
%C A106396 1, 2, 1;
%C A106396 1, 4, 2, 1;
%C A106396 1, 7, 5, 2, 1;
%C A106396 1, 11, 13, 5, 2, 1;
%C A106396 1, 16, 31, 14, 5, 2, 1;
%C A106396 ...
%C A106396 Second column = A000124.
%F A106396 n-th column (offset) is generated by P * V; P = the Narayana triangle as an infinite lower triangular matrix, V = vector for n-th column comprised of n leading 1's and the rest zeros (e.g. V for 3rd column = [1, 1, 1, 0, 0, 0...].
%e A106396 Col. 2 offset = 1, 2, 4, 7, 11, 16, 22...since P * [1, 1, 0, 0, 0...] = 1, 2, 4, 7, 11...
%Y A106396 Cf. A000124, A001263.
%Y A106396 Adjacent sequences: A106393 A106394 A106395 this_sequence A106397 A106398 A106399
%Y A106396 Sequence in context: A092479 A124022 A098063 this_sequence A048004 A114394 A059623
%K A106396 nonn,tabl
%O A106396 1,5
%A A106396 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2005

%I A048004
%S A048004 1,1,1,1,2,1,1,4,2,1,1,7,5,2,1,1,12,11,5,2,1,1,20,23,12,5,2,1,1,33,47,27,
%T A048004 12,5,2,1,1,54,94,59,28,12,5,2,1,1,88,185,127,63,28,12,5,2,1,1,143,360,
%U A048004 269,139,64,28,12,5,2,1,1,232,694,563,303,143,64,28,12,5,2,1,1,376,1328
%N A048004 Triangular array read by rows: T(n,k)=number of binary words of length n whose longest run of consecutive 1's has length k, for n >= 0, 0 <= k <= n.
%C A048004 Number of compositions of n+1 having largest part equal to k+1. Example: T(4,2)=5 because we have 3+2, 2+3, 3+1+1, 1+3+1, and 1+1+3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
%D A048004 J. Kappraff, Beyond Measure, World Scientific, 2002; see pp. 471-472.
%D A048004 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 155.
%F A048004 T(n, k) = 0 if k < 0 or k > n, 1 if k = 0 or k = n, 2T(n-1, k)+T(n-1, k-1)-2T(n-2, k-1)+T(n-k-1, k-1)-T(n-k-2, k) otherwise - David W. Wilson
%F A048004 T(n, k) = A048887(n+1, k+1)-A048887(n+1, k) - Henry Bottomley (se16(AT)btinternet.com), Oct 29 2002
%F A048004 G.f. for column k=(1-x)^2*x^k/[(1-2x+x^(k+1))*(1-2x+x^(k+2))]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005
%e A048004 Rows: {1}; {1,1}; {1,2,1}; {1,4,2,1}; ...
%e A048004 Example: T(4,2)=5 because we have 0011,0110,1011,1100, and 1101.
%p A048004 G:=k->(1-x)^2*x^k/(1-2*x+x^(k+1))/(1-2*x+x^(k+2)): for k from 0 to 14 do g[k]:=series(G(k),x=0,15) od: 1,seq(seq(coeff(g[k],x^n),k=0..n),n=1..12); (Deutsch)
%Y A048004 See A126198 and A048887 for closely related arrays.
%Y A048004 T(n, 2)=Fibonacci(n+2)-1, A000071, T(n, 3)=b(n) for n=3, 4, ..., where b=A000100, T(n, 4)=c(n) for n=4, 5, ..., where c=A000102.
%Y A048004 Nonnegative elements of columns approach A045623. Cf. A048003.
%Y A048004 Adjacent sequences: A048001 A048002 A048003 this_sequence A048005 A048006 A048007
%Y A048004 Sequence in context: A124022 A098063 A106396 this_sequence A114394 A059623 A057728
%K A048004 nonn,tabl
%O A048004 0,5
%A A048004 Clark Kimberling (ck6(AT)evansville.edu)
%E A048004 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2005

%I A114394
%S A114394 1,1,1,2,1,1,4,2,1,1,8,1,3,2,1,1,16,2,6,4,2,1,1,32,4,12,1,7,4,2,1,1,61,
%T A114394 3,8,24,2,14,8,4,2,1,1,122,3,3,16,48,4,28,1,15,8,4,2,1,1,244,6,3,3,32,
%U A114394 96,8,56,2,30,7,9,8,4,2,1,1,488,12,6,3,3,64,185,7,16,112,4,60,1,13
%N A114394 First differences of A114391.
%Y A114394 Adjacent sequences: A114391 A114392 A114393 this_sequence A114395 A114396 A114397
%Y A114394 Sequence in context: A098063 A106396 A048004 this_sequence A059623 A057728 A098050
%K A114394 nonn
%O A114394 1,4
%A A114394 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Feb 07 2006

%I A059623
%S A059623 1,1,1,2,1,1,4,2,1,1,8,3,2,1,1,15,5,3,2,1,1,27,8,5,3,2,1,1,47,13,7,5,3,
%T A059623 2,1,1,79,21,11,7,5,3,2,1,1,130,33,16,11,7,5,3,2,1,1,209,52,24,15,11,7,
%U A059623 5,3,2,1,1,330,80,35,22,15,11,7,5,3,2,1,1,512,122,52,31,22,15,11,7,5,3
%N A059623 As upper right triangle, number of weakly unimodal partitions of n (weakly unimodal means non-decreasing then non-increasing) where initial part is k (n >= k >= 1).
%F A059623 T(n, k)=S(n, k)-S(n-k, k)+sum_j[T(n-k, j)] for j >= k, where S(n, k)=A008284(n, k)=sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n]=1.
%e A059623 Rows are: {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc. T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.
%Y A059623 Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.
%Y A059623 Adjacent sequences: A059620 A059621 A059622 this_sequence A059624 A059625 A059626
%Y A059623 Sequence in context: A106396 A048004 A114394 this_sequence A057728 A098050 A111579
%K A059623 nonn,tabl
%O A059623 1,4
%A A059623 Henry Bottomley (se16(AT)btinternet.com), Feb 01 2001

%I A057728
%S A057728 1,1,1,1,2,1,1,4,2,1,1,8,4,2,1,1,16,8,4,2,1,1,32,16,8,4,2,1,1,64,32,16,
%T A057728 8,4,2,1,1,128,64,32,16,8,4,2,1,1,256,128,64,32,16,8,4,2,1,1,512,256,
%U A057728 128,64,32,16,8,4,2,1,1,1024,512,256,128,64,32,16,8,4,2,1,1,2048,1024
%N A057728 A triangular table of decreasing powers of two (with first column all ones).
%C A057728 A023758 is the sequence of partial sums of a(n) with row sums A000337. 2^A004736(n) is a sequence closely related to a(n).
%F A057728 First differences of sequence A023758.
%e A057728 When viewed as a triangular array, row 8 of A023758 is 128 192 224 240 248 252 254 255 so row 8 of a(n) is 1 64 32 16 8 4 2 1
%Y A057728 Cf. A000079, A004736, A023758 and A000337.
%Y A057728 Adjacent sequences: A057725 A057726 A057727 this_sequence A057729 A057730 A057731
%Y A057728 Sequence in context: A048004 A114394 A059623 this_sequence A098050 A111579 A122773
%K A057728 base,easy,nonn,tabl
%O A057728 1,5
%A A057728 Alford Arnold (Alford1940(AT)aol.com), Oct 29 2000
%E A057728 More terms from Larry Reeves (larryr(AT)acm.org), Oct 30 2000

%I A098050
%S A098050 1,1,1,1,1,1,2,1,1,4,2,1,1,8,5,2,1,1,16,11,6,2,1,1,32,25,14,7,2,1,1,64,
%T A098050 57,35,17,8,2,1,1,128,130,86,46,20,9,2,1,1,256,296,212,119,58,23,10,2,1,
%U A098050 1,512,672,520,311,156,71,26,11,2,1,1,1024,1520,1269,805,428,197,85,29
%N A098050 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and containing a total of k level steps H in all UHH...HD's, where U=(1,1), H=(1,0) and D=(1,-1) (can be easily expressed using RNA secondary structure terminology).
%C A098050 Row sums yield the RNA secondary structure numbers (A004148).
%D A098050 I. L. Hofacker, P. Schuster, and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
%D A098050 P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
%D A098050 M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%H A098050 M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08viennot.html">Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire</a>, Sem. Loth. Comb. B08l (1984) 79-86.
%F A098050 G.f.=G=G(t, z) satisfies G=1+zG+z^2*G[G-1-z/(1-z)+tz/(1-tz)].
%e A098050 Triangle starts:
%e A098050 1;
%e A098050 1;
%e A098050 1;
%e A098050 1,1;
%e A098050 1,2,1;
%e A098050 1,4,2,1;
%e A098050 1,8,5,2,1;
%e A098050 1,16,11,6,2,1;
%e A098050 Row n has n-1 terms, n>=2.
%e A098050 T(7,3)=5 because we have U(HHH)DHH, HU(HHH)DH, HHU(HHH)D, U(H)DU(HH)D,
%e A098050 U(HH)DU(H)D, and UU(HHH)DD, where U=(1,1), H=(1,0) and D=(1,-1); the
%e A098050 three pertinent H's are shown between parentheses.
%Y A098050 Cf. A004148.
%Y A098050 Adjacent sequences: A098047 A098048 A098049 this_sequence A098051 A098052 A098053
%Y A098050 Sequence in context: A114394 A059623 A057728 this_sequence A111579 A122773 A029268
%K A098050 nonn,tabf
%O A098050 0,7
%A A098050 Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 11 2004

%I A111579
%S A111579 1,1,1,1,2,1,1,4,2,1,1,8,5,2,1,1,16,15,6,2,1,1,32,52,24,7,2,1,1,64,203,
%T A111579 116,35,8,2,1,1,128,877,648,214,48,9,2,1,1,256,4140,4088,1523,352,63,10,
%U A111579 2,1,1,512,21147,28640,12349,3008,536,80,11,2,1
%N A111579 Generalized Bell number triangle, read by rows.
%C A111579 Generalized Stirling number of the second kind triangles may be defined by the generating operation T(n,k) = T(n-1,k-1) + Q*T(n-1,k) where Q denotes an arithmetic sequence (1,1,1...Pascal's triangle); (1,2,3...Stirling number of the second kind triangle); (1,3,5...A039755 an analogue of the Stirling number of the second kind triangle); etc...
%F A111579 Columns are row sums of generalized Stirling number of the second kind triangles.
%e A111579 Column 2 (1, 2, 5, 15, 52, 203...are Bell numbers deleting the first 1); which are row sums of the Stirling number of the second kind triangle A008277.
%e A111579 Column 3 (1, 2, 6, 24, 116...) = row sums of A039755, a Stirling number of the second kind analogue.
%Y A111579 Cf. A008277, A000110, A039755, A004211, A111577, A111578.
%Y A111579 Adjacent sequences: A111576 A111577 A111578 this_sequence A111580 A111581 A111582
%Y A111579 Sequence in context: A059623 A057728 A098050 this_sequence A122773 A029268 A064191
%K A111579 nonn,tabl,uned
%O A111579 0,5
%A A111579 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2005

%I A122773
%S A122773 2,1,1,4,2,1,4,4,2,1,16,24,16,4,1,16,32,28,12,3,1,64,160,176,104,36,6,1,64,
%T A122773 192,256,192,88,24,4,1,256,896,1408,1280,736,272,64,8,1,256,1024,1856,1984,
%U A122773 1376,640,200,40,5,1,1024,4608,9472,11648,9472,5312,2080,560,100,10,1,1024
%V A122773 2,1,-1,-4,2,1,4,-4,2,-1,-16,24,-16,4,1,16,-32,28,-12,3,-1,-64,160,-176,104,-36,6,1,64,
%W A122773 -192,256,-192,88,-24,4,-1,-256,896,-1408,1280,-736,272,-64,8,1,256,-1024,1856,-1984,
%X A122773 1376,-640,200,-40,5,-1,-1024,4608,-9472,11648,-9472,5312,-2080,560,-100,10,1,1024
%N A122773 Triangular array from Bonacci type "field": matrices: 2*(I+A[i,j]^(-1))^(-1)=Sum[A[i,j]^n,{n,0,Infinity}] Characteristic Polynomials: 2, 1 - x, -4 + 2 x + x^2, 4 - 4 x + 2 x^2 - x^3, -16 + 24 x - 16 x^2 + 4 x^3 + x^4, 16 - 32 x + 28 x^2 - 12 x^3 + 3 x^4 - x^5.
%C A122773 It is necessary to multiply by 2 to get a repeating 1/2 factor out. 2 X 2: {{-2, 2}, {2, 0}} 3 X 3: {{1, 2, -1}, {-1, 0, 1}, {1, 0, 1}} 4 X 4: {{-2, 2, -4, 2}, {2, 0, 4, -2}, {-2, 0, -2, 2}, {2, 0, 2, 0}}, 5 X 5: {{1, 2, -1, 2, -1}, {-1, 0, 1, -2, 1}, {1, 0, 1, 2, -1}, {-1, 0, -1, 0, 1}, {1, 0, 1, 0, 1}}, 6 X 6: {{-2, 2, -4, 2, -4, 2}, {2, 0, 4, -2, 4, -2}, {-2, 0, -2, 2, -4, 2}, {2, 0, 2, 0, 4, -2}, {-2, 0, -2, 0, -2, 2}, {2, 0, 2, 0, 2, 0}}
%D A122773 Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
%D A122773 Kappraff, J., Blackmore, D., and Adamson, G. "Phyllotaxis as a Dynamical System: A Study in Number." In Symmetry in Plants edited by R.V. Jean and D. Barabe. Singapore: World Scientific. (1996).
%D A122773 Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%F A122773 a(i,j)=(I[n]+M(i,j)^(-1))^(-1) 2*a(i,j)->p(m,x) p(n,x)->t(n,m)
%e A122773 Triangular array:
%e A122773 {2},
%e A122773 {1, -1},
%e A122773 {-4, 2, 1},
%e A122773 {4, -4, 2, -1},
%e A122773 {-16, 24, -16, 4, 1},
%e A122773 {16, -32, 28, -12, 3, -1},
%e A122773 {-64, 160, -176, 104, -36, 6, 1},
%e A122773 {64, -192, 256, -192, 88, -24, 4, -1}
%t A122773 An[d_] := Table[If[n == d, 1, If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}]; Join[{{2}}, Table[CoefficientList[CharacteristicPolynomial[2*IdentityMatrix[d] + MatrixPower[An[d], -1], x], x], {d, 1, 20}]] Flatten[%]
%Y A122773 Adjacent sequences: A122770 A122771 A122772 this_sequence A122774 A122775 A122776
%Y A122773 Sequence in context: A057728 A098050 A111579 this_sequence A029268 A064191 A127420
%K A122773 uned,tabl,sign
%O A122773 1,1
%A A122773 Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 20 2006

%I A029268
%S A029268 1,0,0,1,1,0,1,1,1,2,1,1,4,2,1,4,4,2,5,4,4,7,5,4,10,7,5,
%T A029268 11,10,7,13,11,10,16,13,11,21,16,13,23,21,16,26,23,21,31,
%U A029268 26,23,38,31,26,41,38,31,46,41,38,53,46,41,62,53,46,67
%N A029268 Expansion of 1/((1-x^3)(1-x^4)(1-x^9)(1-x^12)).
%Y A029268 Adjacent sequences: A029265 A029266 A029267 this_sequence A029269 A029270 A029271
%Y A029268 Sequence in context: A098050 A111579 A122773 this_sequence A064191 A127420 A129033
%K A029268 nonn
%O A029268 0,10
%A A029268 njas

%I A064191
%S A064191 1,1,1,2,1,1,4,2,2,1,9,4,5,2,1,21,9,12,5,3,1,51,21,30,12,9,3,1,127,51,
%T A064191 76,30,25,9,4,1,323,127,196,76,69,25,14,4,1,835,323,512,196,189,69,44,
%U A064191 14,5,1,2188,835,1353,512,518,189,133,44,20,5,1,5798,2188,3610,1353
%N A064191 Triangle T(n,k) (n >= 0, 0 <= k <= n) generalizing Motzkin numbers.
%C A064191 This triangle appears on page 9 of the linked reference, and is defined by Corollary 2.4.
%H A064191 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles.
%F A064191 T(n, 0) = sum(T(n-1, k) : k = 0, ..., n-1). For k even, 0 < k <= n, T(n, k) = sum(T(n-1, j) : j = k-1, ..., n-1). For k odd, 0 < k <= n, T(n, k) = T(n-1, k-1). - David Wasserman (wasserma(AT)spawar.navy.mil), Jul 15 2002
%e A064191 1; 1,1; 2,1,1; 4,2,2,1; ...
%Y A064191 First column gives A001006.
%Y A064191 Adjacent sequences: A064188 A064189 A064190 this_sequence A064192 A064193 A064194
%Y A064191 Sequence in context: A111579 A122773 A029268 this_sequence A127420 A129033 A054090
%K A064191 nonn,tabl,easy
%O A064191 0,4
%A A064191 njas, Sep 21 2001
%E A064191 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jul 15 2002

%I A127420
%S A127420 1,1,1,2,1,1,4,2,2,1,9,5,5,3,1,1,24,15,15,10,5,5,2,1,77,53,53,38,23,23,
%T A127420 13,8,3,3,1,295,218,218,165,112,112,74,51,28,28,15,7,4,1,1,1329,1034,
%U A127420 1034,816,598,598,433,321,209,209,135,84,56,28,28,13,6,2,1,6934,5605
%N A127420 Triangle, read by rows, where row n+1 is generated from row n by first inserting zeros at positions {(m+2)*(m+3)/2, m>=0} in row n and then taking the partial sums in reverse order, for n>=2, starting with 1's in the initial two rows.
%C A127420 Column 0 forms A091352, which also equals column 1 of table A125781, where table A125781 is generated by a complementary recurrence of this triangle. The number of terms in row n is A127419(n).
%e A127420 To generate row 6, start with row 5:
%e A127420 24, 15, 15, 10, 5, 5, 2, 1;
%e A127420 insert zeros at positions [1,4,8,13,..., (m+2)*(m+3)/2 - 2,...]:
%e A127420 24, 0, 15, 15, 0, 10, 5, 5, 0, 2, 1;
%e A127420 then row 6 equals the partial sums of row 5 taken in reverse order:
%e A127420 24, _0, 15, 15, _0, 10, _5, 5, 0, 2, 1;
%e A127420 77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1.
%e A127420 Triangle begins:
%e A127420 1;
%e A127420 1, 1;
%e A127420 2, 1, 1;
%e A127420 4, 2, 2, 1;
%e A127420 9, 5, 5, 3, 1, 1;
%e A127420 24, 15, 15, 10, 5, 5, 2, 1;
%e A127420 77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1;
%e A127420 295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1;
%e A127420 1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1;
%e A127420 Column 0 of this triangle equals column 1 of triangle A091351, where triangle A091351 begins:
%e A127420 1;
%e A127420 1, 1;
%e A127420 1, 2, 1;
%e A127420 1, 4, 3, 1;
%e A127420 1, 9, 9, 4, 1;
%e A127420 1, 24, 30, 16, 5, 1;
%e A127420 1, 77, 115, 70, 25, 6, 1;
%e A127420 1, 295, 510, 344, 135, 36, 7, 1; ...
%e A127420 and column k of A091351 = row sums of matrix power A091351^k for k>=0.
%Y A127420 Cf. A091352, A091351, A125781, A127419.
%Y A127420 Adjacent sequences: A127417 A127418 A127419 this_sequence A127421 A127422 A127423
%Y A127420 Sequence in context: A122773 A029268 A064191 this_sequence A129033 A054090 A122517
%K A127420 nonn,tabl
%O A127420 0,4
%A A127420 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 14 2007

%I A129033
%S A129033 0,1,1,2,1,1,4,2,2,4,5,2
%N A129033 Number of n-node triangulations of the torus S_1 in which every node has degree >= 6.
%D A129033 M. Jungerman and G. Ringel, Minimal triangulations on orientable surfaces, Acta Math. 145 (1980), 121-154.
%H A129033 Thom Sulanke, <a href="http://hep.physics.indiana.edu/~tsulanke/graphs/surftri/">Generating triangulations of surfaces (surftri)</a>, (also subpages).
%Y A129033 Adjacent sequences: A129030 A129031 A129032 this_sequence A129034 A129035 A129036
%Y A129033 Sequence in context: A029268 A064191 A127420 this_sequence A054090 A122517 A123199
%K A129033 nonn
%O A129033 6,4
%A A129033 njas, May 12 2007

%I A054090
%S A054090 1,1,1,1,2,1,1,4,2,3,1,10,6,8,7,1,32,22,26,24,25,1,130,98,108,104,106,
%T A054090 105,1,652,522,554,544,548,546,547,1,3914,3262,3392,3360,3370,3366,3368,
%U A054090 3367,1,27400,23486,24138,24008,24040,24030,24034,24032,24033,1
%N A054090 Triangular array generated by its row sums: T(n,0)=1 for n >= 0, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+d*r(n-k), d=(-1)^(k+1), for k=2