The Database of Integer Sequences, Part 11 

     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 A089329
%S A089329 1,2,1,7,2,2,24,8,1,103,1,12,43,94,21,1,11,12,4,23,27,4,89,20,42,1,43,6,
%T A089329 41,44,190,22,12,139,41,114,16,3,26,171,32,220,78,86,135,117,21,44,49,
%U A089329 143,248,175,9,76,6,66,426,46,237,252,9,62,319,88,150,123,61,122,300,15
%N A089329 Smallest k such that the concatenation r*k for r = 1 to n followed by a 1 is a prime.
%e A089329 a(4) = 7 and the prime is 71421281= A089328(4).
%t A089329 Do[s = ""; k = 0; While[ !PrimeQ[ToExpression[s]], s = ""; k++; For[r = 1, r <= n, r++, s = s <> ToString[r*k]]; s = s <> "1"]; Print[k], {n, 1, 50}] (Propper)
%Y A089329 Cf. A089328.
%Y A089329 Adjacent sequences: A089326 A089327 A089328 this_sequence A089330 A089331 A089332
%Y A089329 Sequence in context: A128747 A124392 A121416 this_sequence A097411 A134929 A136535
%K A089329 base,nonn
%O A089329 1,2
%A A089329 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 04 2003
%E A089329 Corrected and extended by Ryan Propper (rpropper(AT)stanford.edu), Jul 24 2005
%E A089329 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Sep 13 2005

%I A097411
%S A097411 1,2,1,7,2,8,2,4,1,1,3,6,1,1,2,4,6,8,1,1,2,3,4,6,8,1,1,1,2,2,3,4,5,6,7,
%T A097411 9,1,1,1,1,2,2,3,3,4,5,5,6,7,8,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5,6,6,7,8,9,
%U A097411 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,5,5,6,6,6,7,8,8,9,1,1,1,1
%N A097411 Initial decimal digit of n^7.
%H A097411 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GelfandsQuestion.html">Gelfand's Question</a>
%e A097411 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, ...
%Y A097411 Cf. A000027, A002993, A002994, A097408, A097409, A097410, A097412, A097413, A097414.
%Y A097411 Adjacent sequences: A097408 A097409 A097410 this_sequence A097412 A097413 A097414
%Y A097411 Sequence in context: A124392 A121416 A089329 this_sequence A134929 A136535 A091370
%K A097411 nonn,base,easy
%O A097411 1,2
%A A097411 Eric Weisstein (eric(AT)weisstein.com), Aug 16, 2004

%I A134929
%S A134929 2,1,7,2,9,8,7,5,3,9,3,1,9,6,9,6,3,4,7,2,3,0,9,2,4,8,4,8,9,8,8,6,5,9,2,
%T A134929 7,6,9,6,2,4,9,6
%N A134929 Successive digits of taxicab numbers A011541(n).
%Y A134929 Adjacent sequences: A134926 A134927 A134928 this_sequence A134930 A134931 A134932
%Y A134929 Sequence in context: A121416 A089329 A097411 this_sequence A136535 A091370 A125697
%K A134929 base,easy,nonn
%O A134929 1,1
%A A134929 Omar E. Pol (info(AT)polprimos.com), Nov 17 2007

%I A136535
%S A136535 1,1,2,1,7,3,1,15,21,4,1,26,76,46,5,1,40,20,250,85,6,1,57,435,925,645,
%T A136535 141,7,1,77,833,2695,3185,1421,217,8,1,100,1456,6664,11956,9016,2800,
%U A136535 316,9,1,126,2376,14616,37044,42336,22176,5076,441,10
%N A136535 A128064 * A001263.
%C A136535 Row sums = A076540: (1, 3, 11, 41, 154, 582, 2211,...).
%F A136535 A001263 = the Narayana triangle.
%e A136535 First few rows of the triangle are:
%e A136535 1;
%e A136535 1, 2;
%e A136535 1, 7, 3;
%e A136535 1, 15, 21, 4;
%e A136535 1, 26, 76, 46, 5;
%e A136535 1, 40, 200, 250, 85, 6;
%e A136535 1, 57, 435, 925, 645, 141, 7;
%e A136535 ...
%Y A136535 Cf. A001263, A128064, A076540.
%Y A136535 Adjacent sequences: A136532 A136533 A136534 this_sequence A136536 A136537 A136538
%Y A136535 Sequence in context: A089329 A097411 A134929 this_sequence A091370 A125697 A090699
%K A136535 nonn,tabl
%O A136535 1,3
%A A136535 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 04 2008

%I A091370
%S A091370 1,2,1,7,3,1,28,12,4,1,121,52,18,5,1,550,237,84,25,6,1,2591,1119,403,
%T A091370 125,33,7,1,12536,5424,1976,630,176,42,8,1,61921,26832,9860,3206,930,
%U A091370 238,52,9,1,310954,134913,49912,16470,4908,1316,312,63,10,1,1582791
%N A091370 Triangle read by rows: T(n,k)=number of dissections of a convex n-gon by nonintersecting diagonals, having a k-gon over a fixed edge (base).
%C A091370 Row sums give the little Schroeder numbers (A001003). Column 3 (first column, corresponding to k=3) gives A010683.
%C A091370 Number of short bushes (i.e. ordered trees with no vertices of outdegree 1) with n-1 leaves and having root of degree k-1. Example: T(5,3)=7 because, in addition to the five binary trees with 6 edges we have (i) two edges rb, rc hanging from the root r with three edges hanging from vertex b and (ii) two edges rb, rc hanging from the root r with three edges hanging from vertex c.
%D A091370 P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.
%H A091370 J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/abs/math.CO/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>
%F A091370 T(n, k)=[(k-1)/(n-k)]sum(2^j*binomial(n-2, n-k-1-j)*binomial(n-k, j), j=0..n-k-1). G.f.=t^3*z^3*S^2/(1-tzS), where S = [1+z-sqrt(1-6*z+z^2)]/(4z) is the g.f. of the little Schroeder numbers (A001003).
%e A091370 T(5,4)=3 because the dissections of the pentagon ABCDEA that have a quadrilateral over the base AB are obtained by the diagonals (i) CE, (ii) AD, and (iii) BD, respectively.
%e A091370 1; 2,1; 7,3,1; 28,12,4,1; 121,52,18,5,1;
%p A091370 a := proc(n,k) if k=0 or k=1 or k=2 then 0 elif k=n then 1 elif k<n then (k-1)*sum(2^j*binomial(n-2,n-k-1-j)*binomial(n-k,j),j=0..n-k-1)/(n-k) else 0 fi end:seq(seq(a(n,k),k=3..n),n=3..13);
%Y A091370 Cf. A001003, A010683.
%Y A091370 Adjacent sequences: A091367 A091368 A091369 this_sequence A091371 A091372 A091373
%Y A091370 Sequence in context: A097411 A134929 A136535 this_sequence A125697 A090699 A120903
%K A091370 nonn,tabl
%O A091370 3,2
%A A091370 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004

%I A125697
%S A125697 1,2,1,7,3,1,35,16,3,1,228,77,20,3,1,2237,485,111,21,3,1,31559
%N A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.
%C A125697 This is a two-dimensional Euler transform of A125699.
%H A125697 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Category.html">Category</a>
%F A125697 G.f. Product_{i=1}^{infinity} Product_{j=1}^{ceiling(i/2)} 1 / (1 - x^i y^j)^A125699(i,j).
%F A125697 T(n,k) = A125701(n-k) when k >= 2/3 n.
%e A125697 The table starts:
%e A125697 1
%e A125697 2,1
%e A125697 7,3,1
%e A125697 35,16,3,1
%Y A125697 Cf. A125696 (row sums), A058129 (column 1), A125699, A125701, A125726.
%Y A125697 Adjacent sequences: A125694 A125695 A125696 this_sequence A125698 A125699 A125700
%Y A125697 Sequence in context: A134929 A136535 A091370 this_sequence A090699 A120903 A021050
%K A125697 tabl,hard,more,nonn
%O A125697 1,2
%A A125697 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) and Christian Bower (bowerc(AT)usa.net), Jan 05 2007

%I A090699
%S A090699 2,1,7,3,2,5,4,3,1,2,5,1,9,5,5,4,1,3,8,2,3,7,0,8,9,8,4,0,4,3,8,2,2,3,7,
%T A090699 2,2,9,0,6,7,1,1,3,2,9,1,3,1,6,6,0,8,5,6,7,4,9,1,7,5,7,5,8,9,6,7,0,5,9,
%U A090699 6,6,1,7,2,6,6,4,4,4,6,8,2,0,3,7,8,5,7,2,7,8,3,8,3,1,7,6,5,1,0,2,6,6,4
%N A090699 Decimal expansion of the Erdos-Szekeres constant zeta(3/2)/zeta(3).
%C A090699 Let N(x) denotes the number of 2-full integers not exceeding x. Then limit x ->infty N(x)/sqrt(x)=zeta(3/2)/zeta(3). Also related to Niven's constant.
%D A090699 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 112-114.
%D A090699 S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
%F A090699 Product_{p prime} (1+1/p^(3/2)) = zeta(3/2)/zeta(3) - T. D. Noe (noe(AT)sspectra.com), May 03 2006
%e A090699 zeta(3/2)/zeta(3) = 2.17325431251955413823708984...
%t A090699 RealDigits[N[Zeta[3/2]/Zeta[3],150]] - T. D. Noe (noe(AT)sspectra.com), May 03 2006
%Y A090699 Cf. A001694 (powerful numbers), A102834 (non-square powerful numbers).
%Y A090699 Cf. A033150.
%Y A090699 Adjacent sequences: A090696 A090697 A090698 this_sequence A090700 A090701 A090702
%Y A090699 Sequence in context: A136535 A091370 A125697 this_sequence A120903 A021050 A115629
%K A090699 cons,nonn
%O A090699 1,1
%A A090699 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 14 2004
%E A090699 Edited by njas at the suggestion of Andrew Plewe, May 16 2007

%I A120903
%S A120903 1,1,1,1,2,1,7,3,3,1,61,28,6,4,1,751,305,70,10,5,1,11821,4506,915,140,15,6,1,
%T A120903 226927,82747,15771,2135,245,21,7,1,5142061,1815416,330988,42056,4270,392,28,8,
%U A120903 1,134341711,46278549,8169372,992964,94626,7686,588,36,9,1,3975839341,1343417110
%V A120903 1,1,1,-1,2,1,7,-3,3,1,-61,28,-6,4,1,751,-305,70,-10,5,1,-11821,4506,-915,140,-15,6,1,
%W A120903 226927,-82747,15771,-2135,245,-21,7,1,-5142061,1815416,-330988,42056,-4270,392,-28,8,
%X A120903 1,134341711,-46278549,8169372,-992964,94626,-7686,588,-36,9,1,-3975839341,1343417110
%N A120903 Triangle T, read by rows, that satisfies matrix equation: T + (T-I)^2 = C, where C is Pascal's triangle.
%C A120903 Column 0 is signed A048287, which is the number of semiorders on n labeled nodes.
%F A120903 E.g.f. A(x,y) satisfies: A(x,y) + [A(x,y) - exp(x*y)]^2 = exp(x+x*y); explicity, e.g.f: A(x,y) = exp(x*y)*(1 + sqrt(4*exp(x)-3))/2. E.g.f. of column 0: (1 + sqrt(4*exp(x)-3))/2. T(n,k) = -(-1)^(n-k)*A048287(n-k)*C(n,k) + 2*0^(n-k). Matrix square: [T^2](n,k) = ( C(n,k) + 2*T(n,k) - 0^(n-k) )/2.
%e A120903 Triangle T begins:
%e A120903 1;
%e A120903 1, 1;
%e A120903 -1, 2, 1;
%e A120903 7, -3, 3, 1;
%e A120903 -61, 28, -6, 4, 1;
%e A120903 751, -305, 70, -10, 5, 1;
%e A120903 -11821, 4506, -915, 140, -15, 6, 1;
%e A120903 226927, -82747, 15771, -2135, 245, -21, 7, 1;
%e A120903 -5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1;
%e A120903 The matrix square of T less the diagonal is (T-I)^2:
%e A120903 0;
%e A120903 0, 0;
%e A120903 2, 0, 0;
%e A120903 -6, 6, 0, 0;
%e A120903 62, -24, 12, 0, 0;
%e A120903 -750, 310, -60, 20, 0, 0;
%e A120903 11822, -4500, 930, -120, 30, 0, 0;
%e A120903 where C = T + (T-I)^2 = 2*T^2 - 2*T + I.
%o A120903 (PARI) /* Generated by Recursion T = C - (T-I)^2 : */ {T(n, k)=local(C=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1))), M=C); for(i=1, n+1, M=C-(M-M^0)^2 ); return(M[n+1, k+1])} (PARI) /* Generated by E.G.F.: */ {T(n,k)=n!*polcoeff(polcoeff(exp(x*y)*(1 + sqrt(4*exp(x +x*O(x^n))-3))/2,n,x),k,y)}
%Y A120903 Cf. A048287 (column 0); A117269 (variant: T - (T-I)^2 = C).
%Y A120903 Adjacent sequences: A120900 A120901 A120902 this_sequence A120904 A120905 A120906
%Y A120903 Sequence in context: A091370 A125697 A090699 this_sequence A021050 A115629 A072248
%K A120903 sign,tabl
%O A120903 0,5
%A A120903 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006

%I A021050
%S A021050 0,2,1,7,3,9,1,3,0,4,3,4,7,8,2,6,0,8,6,9,5,6,5,2,1,7,3,9,1,3,0,4,3,
%T A021050 4,7,8,2,6,0,8,6,9,5,6,5,2,1,7,3,9,1,3,0,4,3,4,7,8,2,6,0,8,6,9,5,6,
%U A021050 5,2,1,7,3,9,1,3,0,4,3,4,7,8,2,6,0,8,6,9,5,6,5,2,1,7,3,9,1,3,0,4,3
%N A021050 Decimal expansion of 1/46.
%Y A021050 Adjacent sequences: A021047 A021048 A021049 this_sequence A021051 A021052 A021053
%Y A021050 Sequence in context: A125697 A090699 A120903 this_sequence A115629 A072248 A092276
%K A021050 nonn,cons
%O A021050 0,2
%A A021050 njas

%I A115629
%S A115629 2,1,7,3,29,30,31,4,8,9,10,5,121,122,123,6,16,11,32,33,34,35,36,12,17,
%T A115629 13,497,498,499,500,501,14,18,15,67,37,46,38,124,19,20,125,126,127,128,
%U A115629 129,130,21,47,39,68,40,48,41,2017,22,2018,2019,2020,2021,2022,2023
%N A115629 Inverse permutation to sequence A113820.
%e A115629 A113820(29) = 5, so a(5) = 29.
%Y A115629 Cf. A113820.
%Y A115629 Adjacent sequences: A115626 A115627 A115628 this_sequence A115630 A115631 A115632
%Y A115629 Sequence in context: A090699 A120903 A021050 this_sequence A072248 A092276 A011274
%K A115629 nonn
%O A115629 1,1
%A A115629 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 27 2006

%I A072248
%S A072248 1,1,2,1,7,4,1,20,26,8,1,54,126,76,16,1,143,548,504,200,32,1,376,2259,
%T A072248 2900,1656,496,64,1,986,9034,15506,11528,4896,1184,128,1,2583,35469,
%U A072248 79354,73172,39552,13536,2752,256,1,6764,137644,394642,439272,285992
%N A072248 Triangle T(n,k) (n>=2, 1<=k<=n-1) giving number of non-crossing trees with n nodes and height k.
%C A072248 For n>2 n-th row has n-2 terms.
%D A072248 E. Deutsch and M. Noy, Statistics on non-crossing trees, Discr. Math., 254 (2002), 75-87.
%F A072248 Column g.f. are T[k]-T[k-1] (k=1, 2, ...), where T[0]=z and T[k]=z/[1-T[k-1]^2/z]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 30 2004
%e A072248 1; 1,2; 1,7,4; 1,20,26,8; ...
%p A072248 T[0]:=z: for k from 1 to 10 do T[k]:=simplify(z/(1-T[k-1]^2/z)) od:for k from 1 to 10 do t[k]:=series(T[k]-T[k-1],z=0,15) od: for n from 1 to 10 do seq(coeff(t[k],z^(n+1)),k=1..n) od; (Deutsch)
%Y A072248 Cf. A001764, A072247. Row sums give A001764.
%Y A072248 Adjacent sequences: A072245 A072246 A072247 this_sequence A072249 A072250 A072251
%Y A072248 Sequence in context: A120903 A021050 A115629 this_sequence A092276 A011274 A122843
%K A072248 nonn,tabl
%O A072248 0,3
%A A072248 njas, Jul 06 2002
%E A072248 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 30 2004

%I A092276
%S A092276 1,2,1,7,4,1,30,18,6,1,143,88,33,8,1,728,455,182,52,10,1,3876,2448,1020,
%T A092276 320,75,12,1,21318,13566,5814,1938,510,102,14,1,120175,76912,33649,
%U A092276 11704,3325,760,133,16,1,690690,444015,197340,70840,21252,5313,1078,168
%N A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.
%D A092276 P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math. 204 (1999), 203-229.
%D A092276 M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
%F A092276 T(n, k)=2k*binomial(3n-k, n-k)/(3n-k). G.f. = 1/(1-tzg^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
%F A092276 T(n, k) = Sum_{j, j>=1} j*T(n-1, k-2+j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
%e A092276 1; 2,1; 7,4,1; 30,18,6,1; 143,88,33,8,1;
%p A092276 T := proc(n,k) if k=n then 1 else 2*k*binomial(3*n-k,n-k)/(3*n-k) fi end: seq(seq(T(n,k),k=1..n),n=1..11);
%Y A092276 Row sums give sequence A001764.
%Y A092276 First column gives sequence A006013.
%Y A092276 Adjacent sequences: A092273 A092274 A092275 this_sequence A092277 A092278 A092279
%Y A092276 Sequence in context: A021050 A115629 A072248 this_sequence A011274 A122843 A107865
%K A092276 nonn,tabl
%O A092276 1,2
%A A092276 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 24 2004

%I A011274
%S A011274 1,2,1,7,4,1,31,18,6,1,154,90,33,8,1,820,481,185,52,10,1
%N A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).
%D A011274 J.M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50, 1994, 135-145.
%H A011274 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%Y A011274 Cf. A011270, A011272.
%Y A011274 Adjacent sequences: A011271 A011272 A011273 this_sequence A011275 A011276 A011277
%Y A011274 Sequence in context: A115629 A072248 A092276 this_sequence A122843 A107865 A089225
%K A011274 nonn,easy,tabl,nice
%O A011274 1,2
%A A011274 pallo(AT)u-bourgogne.fr (Jean Pallo)

%I A122843
%S A122843 1,2,1,7,4,1,32,21,6,1,180,130,41,8,1,1200,930,312,67,10,1,9240,7560,
%T A122843 2646,602,99,12,1,80640,68880,24864,5880,1024,137,14,1,786240,695520,
%U A122843 257040,62496,11304,1602,181,16,1
%N A122843 Triangle read by rows: T[n,k] = the number of ascending runs of length k in the permutations of [n] for k <= n.
%F A122843 T[n,k] = n![(n(k(k+1)-1) - k(k-2)(k+2) + 1]/(k+2)! for 0<k<n; T[n,n] = 1; T[n,k] = A122844(n,k) - A122844(n,k+1)
%e A122843 1
%e A122843 2 1
%e A122843 7 4 1 ; there are 4 ascending runs of length 2 in the permutations of [3], namely 13 in 132 and in 213, 23 in 231, 12 in 312. T[3,2] = 4.
%Y A122843 Cf. A122844, A001710, A006157, A005460.
%Y A122843 Adjacent sequences: A122840 A122841 A122842 this_sequence A122844 A122845 A122846
%Y A122843 Sequence in context: A072248 A092276 A011274 this_sequence A107865 A089225 A075085
%K A122843 easy,nonn,tabl
%O A122843 1,2
%A A122843 David J. Scambler (dscambler(AT)bmm.com), Sep 13 2006

%I A107865
%S A107865 1,1,1,1,2,1,7,4,3,1,77,26,9,4,1,1145,287,67,16,5,1,21410,4412,798,139,
%T A107865 25,6,1,481683,86004,13029,1830,251,36,7,1,12655196,2017658,268368,
%U A107865 32191,3667,412,49,8,1,379998938,55134458,6630228,705680,69868,6657,631
%V A107865 1,-1,1,-1,-2,1,-7,-4,-3,1,-77,-26,-9,-4,1,-1145,-287,-67,-16,-5,1,-21410,-4412,-798,
%W A107865 -139,-25,-6,1,-481683,-86004,-13029,-1830,-251,-36,-7,1,-12655196,-2017658,-268368,
%X A107865 -32191,-3667,-412,-49,-8,1,-379998938,-55134458,-6630228,-705680,-69868,-6657,-631
%N A107865 Matrix inverse of triangle A107862.
%C A107865 Matrix product with A107867 equals A107876.
%e A107865 Triangle begins:
%e A107865 1;
%e A107865 -1,1;
%e A107865 -1,-2,1;
%e A107865 -7,-4,-3,1;
%e A107865 -77,-26,-9,-4,1;
%e A107865 -1145,-287,-67,-16,-5,1;
%e A107865 -21410,-4412,-798,-139,-25,-6,1;
%e A107865 -481683,-86004,-13029,-1830,-251,-36,-7,1; ...
%o A107865 (PARI) {T(n,k)=(matrix(n+1,n+1,r,c,if(r>=c, binomial((r-1)*(r-2)/2-(c-1)*(c-2)/2+r-c,r-c)))^-1)[n+1,k+1]}
%Y A107865 Cf. A107862, A107866 (column 0), A107867, A107870, A107876.
%Y A107865 Adjacent sequences: A107862 A107863 A107864 this_sequence A107866 A107867 A107868
%Y A107865 Sequence in context: A092276 A011274 A122843 this_sequence A089225 A075085 A124048
%K A107865 sign,tabl
%O A107865 0,5
%A A107865 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2005

%I A089225
%S A089225 1,2,1,7,4,3,35,22,17,14,228,154,122,102,88,1834,1310,1060,898,782,694,
%T A089225 17582,13128,10818,9272,8142,7272,6578,195866,151560,126882,109880,
%U A089225 97218,87336,79370,72792,2487832,1981824,1682196,1470304,1309776
%N A089225 Triangle T(n,k) read by rows, defined by T(n,k) = (n-k)*T(n-1,k)+Sum(k=1..n, T(n-1,k)); T(1,1) = 1, T(1,k)= 0 if k >1.
%C A089225 Let M be the n X n matrix with M(i,i)=i, other entries 1. Then T(n,k) = permanent of n-1 X n-1 matrix obtained by omitting row k and column k from M.
%C A089225 T(n,1) = A003713(n). n-th row sum = T(n+1,n+1) = A007840(n). {1}, {2, 1}, {7, 4, 3}, {35, 22, 17, 14}, ...
%e A089225 n=4: M = |1,1,1,1|1, 2,1, 1|1, 1, 3, 1|1, 1, 1, 4|
%e A089225 T(4, 1) = permanent of |2, 1, 1|1, 3, 1|1, 1, 4| = 26+5+4 = 35
%e A089225 T(4, 2) = permanent of |1, 1, 1|1, 3, 1|1, 1, 4| = 13+5+4 = 22
%e A089225 T(4, 3) = permanent of |1, 1, 1|1, 2, 1|1, 1, 4| = 9+5+3 = 17
%e A089225 T(4, 4) = permanent of |1, 1, 1|1, 2, 1|1, 1, 3| = 7+4+3 = 14
%Y A089225 Adjacent sequences: A089222 A089223 A089224 this_sequence A089226 A089227 A089228
%Y A089225 Sequence in context: A011274 A122843 A107865 this_sequence A075085 A124048 A087059
%K A089225 easy,nonn,tabl
%O A089225 1,2
%A A089225 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 10 2003

%I A075085
%S A075085 2,1,7,4,8,17,12,6,35,24,39,30,20,10,67,36,95,14,42,28,87,48,32,137,72,
%T A075085 238,22,44,131,161,55,179,78,26,130,177,84,247,60,90,269,213,170,34,68,
%U A075085 233,5,204,295,265,76,114,38,190,371,120,389,313,132,88,327,230,15,399
%N A075085 Rearrangement of natural numbers such that the n-th partial sum is divisible by the n-th prime.
%Y A075085 Adjacent sequences: A075082 A075083 A075084 this_sequence A075086 A075087 A075088
%Y A075085 Sequence in context: A122843 A107865 A089225 this_sequence A124048 A087059 A120872
%K A075085 nonn
%O A075085 1,1
%A A075085 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 13 2002
%E A075085 More terms from David Wasserman (dwasserm(AT)earthlink.net), Jan 16 2005

%I A124048
%S A124048 1,2,1,7,4,11,8,26,23,30,14,28
%N A124048 a(n) = count of instances where the i-th permutation is divisible by i, treating the permutations of {0,1,...n-1} as integers base n in ascending order.
%e A124048 For n = 5, the first, 2nd, 7th and 18th permutations base 3 {01234,01243,02134,03421} = decimal {194,198,294,486} are divisible by 1,2,7 and 18 respectively. a(5) = 4.
%Y A124048 Cf. A124453.
%Y A124048 Adjacent sequences: A124045 A124046 A124047 this_sequence A124049 A124050 A124051
%Y A124048 Sequence in context: A107865 A089225 A075085 this_sequence A087059 A120872 A019642
%K A124048 nonn
%O A124048 1,2
%A A124048 David J. Scambler (dscambler(AT)bmm.com), Nov 02 2006

%I A087059
%S A087059 2,1,7,4,14,9,2,16,7,25,14,1,23,8,34,17,47,28,7,41,18,56,31,4,46,17,63,
%T A087059 32,82,49,14,68,31,89,50,9,71,28,94,49,2,72,23,97,46,124,71,16,98,41,
%U A087059 127,68,7,97,34,128,63,161,94,25,127,56,162,89,14,124,47,161,82,1,119
%N A087059 Difference between 2 * n^2 and the next greater square number.
%F A087059 A087059(n) = A087058(n) - 2*n^2 = A087057(n)^2 - 2*n^2 = (1 + A001951(n))^2 - 2*n^2 = (1 + floor[n*sqrt(2)])^2 - 2*n^2
%e A087059 A087059(10) = 25 because the difference between 2*10^2 = 200 and the next greater square number (225) is 25.
%Y A087059 Cf. A001951, A087055, A087056, A087057, A087058, A087060.
%Y A087059 Adjacent sequences: A087056 A087057 A087058 this_sequence A087060 A087061 A087062
%Y A087059 Sequence in context: A089225 A075085 A124048 this_sequence A120872 A019642 A048505
%K A087059 easy,nonn
%O A087059 1,1
%A A087059 Jens Voss (jens(AT)voss-ahrensburg.de), Aug 07 2003

%I A120872
%S A120872 2,1,7,4,14,9,16,7,25,14,23,8,34,17,47,28,41,18
%N A120872 a(n)=the value of k for row n of the fixed-k dispersion for Q=8.
%C A120872 This sequence results from A087059 by deleting duplicates.
%D A120872 C. Kimberling, The equation (j+k+1)^2-4k=Q*n^2 and related dispersions, preprint.
%e A120872 For each positive integer n, there is a unique pair (j,k) of
%e A120872 positive integers such that (j+k+1)^2-4*k=8*n^2. This
%e A120872 representation is used to define the fixed-k dispersion for Q=8,
%e A120872 given by A120861, having northwest corner
%e A120872 1 7 41 239
%e A120872 2 12 70 408
%e A120872 3 19 111 647
%e A120872 4 24 140 816
%e A120872 The pair (j,k) for each n, shown in the position occupied by
%e A120872 n in the above array, is shown here:
%e A120872 (1,2) (17,2) (43,2) (673,2)
%e A120872 (4,1) (32,1) (196,1) (1152,1)
%e A120872 (2,7) (46,7) (306,7) (1822,7)
%e A120872 (7,4) (63,4) (391,4) (2303,4)
%e A120872 The fixed-k for row 1 is a(1)=2;
%e A120872 the fixed-k for row 2 is a(2)=1; etc.
%e A120872 (For example, (46+7+1)^2-4*7=8*19^2.)
%Y A120872 Cf. A087059, A120861.
%Y A120872 Adjacent sequences: A120869 A120870 A120871 this_sequence A120873 A120874 A120875
%Y A120872 Sequence in context: A075085 A124048 A087059 this_sequence A019642 A048505 A124821
%K A120872 nonn
%O A120872 1,1
%A A120872 Clark Kimberling (ck6(AT)evansville.edu), Jul 10 2006

%I A019642
%S A019642 2,1,7,5,1,2,1,7,6,5,3,2,7,6,2,7,8,6,2,3,5,2,3,1,1,5,1,8,0,5,2,5,5,
%T A019642 4,0,2,6,0,2,3,9,5,8,9,0,3,6,7,9,6,3,6,0,2,7,1,3,5,0,7,9,6,0,7,2,6,
%U A019642 2,0,8,3,1,2,0,2,7,0,2,1,9,4,9,0,7,1,7,1,1,5,9,8,0,2,3,9,7,1,9,0,9
%N A019642 Decimal expansion of sqrt(2*Pi*E)/19.
%Y A019642 Adjacent sequences: A019639 A019640 A019641 this_sequence A019643 A019644 A019645
%Y A019642 Sequence in context: A124048 A087059 A120872 this_sequence A048505 A124821 A104030
%K A019642 nonn,cons
%O A019642 0,1
%A A019642 njas

%I A048505
%S A048505 1,2,1,7,5,1,25,18,10,1,81,56,35,17,1,241,160,101,58,26,1,673,432,269,
%T A048505 160,87,37,1,1793,1120,685,408,233,122,50,1,4609,2816,1693,1000,577,
%U A048505 320,163,65,1,11521,6912,4093,2392,1377,776,421
%N A048505 Array T read by diagonals, n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is (k+n)^2, for n=1,2,3,...; k=0,1,2,...
%F A048505 T(k, n) = (n^2 + (4k+1)n + (2k)^2) * 2^(n-2) - k^2 + 1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Feb 05 2004
%e A048505 Diagonals: {1}; {2,1}; {7,5,1}; ...
%Y A048505 Row 2 = (1, 5, 18, 56, 160, ...) = A001793.
%Y A048505 Adjacent sequences: A048502 A048503 A048504 this_sequence A048506 A048507 A048508
%Y A048505 Sequence in context: A087059 A120872 A019642 this_sequence A124821 A104030 A082791
%K A048505 nonn,tabl
%O A048505 0,2
%A A048505 Clark Kimberling (ck6(AT)evansville.edu)

%I A124821
%S A124821 1,2,1,7,5,1,30,25,8,1,143,130,52,11,1,728,700,320,88,14,1,3876,3876,
%T A124821 1938,627,133,17,1,21318,21945,11704,4235,1078,187,20,1,120175,126500,
%U A124821 70840,27830,8050,1700
%V A124821 1,-2,1,7,-5,1,-30,25,-8,1,143,-130,52,-11,1,-728,700,-320,88,-14,1,3876,-3876,1938,
%W A124821 -627,133,-17,1,-21318,21945,-11704,4235,-1078,187,-20,1,120175,-126500,70840,-27830,
%X A124821 8050,-1700
%N A124821 Number triangle T(n,k)=(-1)^(n-k)*(3k+2)*C(3n+1, n-k)/(2n+k+2).
%C A124821 Inverse of number triangle A124819. Row sums are (-1)^n*A001764(n).
%e A124821 Triangle begins
%e A124821 1,
%e A124821 -2, 1,
%e A124821 7, -5, 1,
%e A124821 -30, 25, -8, 1,
%e A124821 143, -130, 52, -11, 1,
%e A124821 -728, 700, -320, 88, -14, 1,
%e A124821 3876, -3876, 1938, -627, 133, -17, 1
%Y A124821 Cf. A124019.
%Y A124821 Adjacent sequences: A124818 A124819 A124820 this_sequence A124822 A124823 A124824
%Y A124821 Sequence in context: A120872 A019642 A048505 this_sequence A104030 A082791 A077230
%K A124821 easy,sign,tabl
%O A124821 0,2
%A A124821 Paul Barry (pbarry(AT)wit.ie), Nov 08 2006

%I A104030
%S A104030 1,2,1,7,5,1,41,32,9,1,376,299,91,14,1,5033,4015,1241,205,20,1,92821,74080,
%T A104030 22954,3842,400,27,1,2257166,1801537,558402,93652,9863,707,35,1,69981919,
%U A104030 55855829,17313721,2904530,306409,22190,1162,44,1,2694447797,2150565968
%V A104030 1,-2,1,7,-5,1,-41,32,-9,1,376,-299,91,-14,1,-5033,4015,-1241,205,-20,1,92821,-74080,
%W A104030 22954,-3842,400,-27,1,-2257166,1801537,-558402,93652,-9863,707,-35,1,69981919,
%X A104030 -55855829,17313721,-2904530,306409,-22190,1162,-44,1,-2694447797,2150565968
%N A104030 Matrix inverse, read by rows, of triangle A104029, which forms the pair-wise sums of trinomial coefficients.
%C A104030 Column 0 forms signed Hammersley's polynomial p_n(1) (A006846), offset 1. Row sums equal negative Genocchi numbers of first kind (A001469). Rows form polynomials R_n(x) such that: R_n(3) = 1 for n>=0, and R_n(1/2) = (-1)^n*A005647(n+1)/2^n (signed Salie numbers). Column 1 forms A104031. Absolute row sums form A104032.
%e A104030 Rows begin:
%e A104030 1;
%e A104030 -2,1;
%e A104030 7,-5,1;
%e A104030 -41,32,-9,1;
%e A104030 376,-299,91,-14,1;
%e A104030 -5033,4015,-1241,205,-20,1;
%e A104030 92821,-74080,22954,-3842,400,-27,1;
%e A104030 -2257166,1801537,-558402,93652,-9863,707,-35,1; ...
%o A104030 (PARI) {T(n,k)=if(n<k|k<0,0,((matrix(n+2,n+2,m,j,if(m>=j, polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-2)+ polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-1))))^-1)[n+1,k+1])}
%Y A104030 Cf. A006846, A001469, A005647, A104027, A104029, A104031, A104032.
%Y A104030 Adjacent sequences: A104027 A104028 A104029 this_sequence A104031 A104032 A104033
%Y A104030 Sequence in context: A019642 A048505 A124821 this_sequence A082791 A077230 A019668
%K A104030 sign,tabl
%O A104030 0,2
%A A104030 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 26 2005

%I A082791
%S A082791 2,1,7,5,4,4,3,3,3,2,2,2,2,2,14,13,12,12,11,1,1,1,1,1,1,1,1,1,1,7
%N A082791 Smallest k such that k*n begins with 2: a(n) = A082811(n)/n.
%Y A082791 Cf. A082784, A054850, A082786, A082787, A082788, A082789, A082811.
%Y A082791 Adjacent sequences: A082788 A082789 A082790 this_sequence A082792 A082793 A082794
%Y A082791 Sequence in context: A048505 A124821 A104030 this_sequence A077230 A019668 A091700
%K A082791 base,easy,nonn
%O A082791 1,1
%A A082791 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 19 2003

%I A077230
%S A077230 2,1,7,5,787,763,893209,2885597,1153151299,261937547,3997632829,30141297349,
%T A077230 4101190700056349,2948796705108299,320676905674696783,43360062621189833,
%U A077230 5848606947453449297743,1963629536423819469923,575654781675816234791672323
%V A077230 2,1,-7,5,-787,763,-893209,2885597,-1153151299,261937547,-3997632829,30141297349,
%W A077230 -4101190700056349,2948796705108299,-320676905674696783,43360062621189833,
%X A077230 -5848606947453449297743,1963629536423819469923,-575654781675816234791672323
%N A077230 Numerators of coefficients of series expansion of a certain integral in the theory of charged particle beams.
%C A077230 The integral is Integrate[1/Sqrt[Log[y]],{y,1,x}]=Sqrt[Pi]*Erfi[Sqrt[Log[x]] with series expansion Sqrt[x-1]*Sum[c(i)*(x-1)^(i-1),{i,0,19}]. Numerator(c(n))= A077230(n), denominator(c(n))=A077231(n).
%D A077230 M. Reiser, Theory and design of charged particle beams. J. Wiley, N.Y. 1994, S. Humphries, Charged particle beams. J. Wiley, N.Y. 1990.
%e A077230 Series expansion is Sqrt[x-1]*(2 + 1/6 (x-1) -7/240 (x-1)^2+ 5/448 (x-1)^3 -...), hence a(0)=2, a(1)=1, a(2)=-7, a(3)=5, etc.
%Y A077230 Cf. A077231.
%Y A077230 Adjacent sequences: A077227 A077228 A077229 this_sequence A077231 A077232 A077233
%Y A077230 Sequence in context: A124821 A104030 A082791 this_sequence A019668 A091700 A135895
%K A077230 sign,frac
%O A077230 0,1
%A A077230 Zak Seidov (zakseidov(AT)yahoo.com), Oct 31 2002

%I A019668
%S A019668 1,2,1,7,6,1,7,6,5,2,2,1,7,6,1,5,7,7,6,8,9,2,3,4,2,9,4,8,3,7,8,2,0,
%T A019668 0,9,6,4,7,4,9,1,2,8,1,4,6,9,3,6,0,6,9,5,2,4,5,9,0,7,2,5,0,0,4,7,7,
%U A019668 9,9,8,1,7,3,8,4,8,4,5,6,5,3,6,8,5,1,8,3,1,5,2,3,6,5,6,9,6,0,7,8,8
%N A019668 Decimal expansion of sqrt(Pi*E)/24.
%Y A019668 Adjacent sequences: A019665 A019666 A019667 this_sequence A019669 A019670 A019671
%Y A019668 Sequence in context: A104030 A082791 A077230 this_sequence A091700 A135895 A039814
%K A019668 nonn,cons
%O A019668 0,2
%A A019668 njas

%I A091700
%S A091700 1,2,1,7,6,1,23,29,10,1,78,127,67,14,1,264,527,375,121,18,1,895,2113,
%T A091700 1892,831,191,22,1,3034,8269,8922,4973,1559,277,26,1,10286,31781,40115,
%U A091700 27139,10826,2623,379,30,1,34872,120448,174080,138617,67308,20763,4087
%N A091700 Matrix square of triangle A063967.
%e A091700 1; 2,1; 7,6,1; 23,29,10,1; 78,127,67,14,1; ...
%Y A091700 Row sums: A091701. Column 0: A091702.
%Y A091700 Adjacent sequences: A091697 A091698 A091699 this_sequence A091701 A091702 A091703
%Y A091700 Sequence in context: A082791 A077230 A019668 this_sequence A135895 A039814 A078301
%K A091700 nonn,tabl
%O A091700 0,2
%A A091700 Christian G. Bower (bowerc(AT)usa.net), Jan 29 2004

%I A135895
%S A135895 1,2,1,7,6,1,34,39,10,1,215,300,95,14,1,1698,2741,990,175,18,1,16220,
%T A135895 29380,11635,2296,279,22,1,182714,363922,154450,32865,4410,407,26,1,
%U A135895 2378780,5135894,2302142,517916,74319,7524,559,30,1,35219202,81557270
%N A135895 Triangle, read by rows, equal to R^2, the matrix square of R = A135894.
%C A135895 Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.
%F A135895 Column k of R^2 = column 1 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R^2 = column 1 of P; column 1 of R^2 = column 1 of P^3; column 2 of R^2 = column 1 of P^5.
%e A135895 Triangle R^2 begins:
%e A135895 1;
%e A135895 2, 1;
%e A135895 7, 6, 1;
%e A135895 34, 39, 10, 1;
%e A135895 215, 300, 95, 14, 1;
%e A135895 1698, 2741, 990, 175, 18, 1;
%e A135895 16220, 29380, 11635, 2296, 279, 22, 1;
%e A135895 182714, 363922, 154450, 32865, 4410, 407, 26, 1;
%e A135895 2378780, 5135894, 2302142, 517916, 74319, 7524, 559, 30, 1;
%e A135895 35219202, 81557270, 38229214, 8980944, 1353522, 145805, 11830, 735, 34, 1;
%e A135895 where R = A135894 begins:
%e A135895 1;
%e A135895 1, 1;
%e A135895 2, 3, 1;
%e A135895 6, 12, 5, 1;
%e A135895 25, 63, 30, 7, 1;
%e A135895 138, 421, 220, 56, 9, 1;
%e A135895 970, 3472, 1945, 525, 90, 11, 1; ...
%e A135895 where column k of R = column 0 of P^(2k+1)
%e A135895 and P = A135880 begins:
%e A135895 1;
%e A135895 1, 1;
%e A135895 2, 2, 1;
%e A135895 6, 7, 3, 1;
%e A135895 25, 34, 15, 4, 1;
%e A135895 138, 215, 99, 26, 5, 1;
%e A135895 970, 1698, 814, 216, 40, 6, 1; ...
%e A135895 where column k of P equals column 0 of R^(k+1).
%o A135895 (PARI) {T(n,k)=local(P=Mat(1),R=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(R^2)[n+1,k+1]}
%Y A135895 Cf. A135882 (column 0), A135890 (column 1); A135894 (R), A135880 (P), A135888 (P^3), A135892 (P^5).
%Y A135895 Adjacent sequences: A135892 A135893 A135894 this_sequence A135896 A135897 A135898
%Y A135895 Sequence in context: A077230 A019668 A091700 this_sequence A039814 A078301 A060583
%K A135895 nonn,tabl
%O A135895 0,2
%A A135895 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2007

%I A039814
%S A039814 1,2,1,7,6,1,35,40,12,1,228,315,130,20,1,1834,2908,1485,320,
%T A039814 30,1,17582,30989,18508,5005,665,42,1,195866,375611,253400,
%U A039814 81088,13650,1232,56,1,2487832,5112570,3805723,1389612,279048
%V A039814 1,-2,1,7,-6,1,-35,40,-12,1,228,-315,130,-20,1,-1834,2908,-1485,320,
%W A039814 -30,1,17582,-30989,18508,-5005,665,-42,1,-195866,375611,-253400,
%X A039814 81088,-13650,1232,-56,1,2487832,-5112570,3805723,-1389612,279048
%N A039814 Matrix square of Stirling-1 Triangle A008275.
%F A039814 E.g.f. k-th column: ((ln(1+ln(1+x)))^k)/k!.
%e A039814 1; -2,1; 7,-6,1; -35,40,-12,1; ...
%Y A039814 Cf. A039815-A039817. |a(n, 1)| = A003713(n) (first column).
%Y A039814 Adjacent sequences: A039811 A039812 A039813 this_sequence A039815 A039816 A039817
%Y A039814 Sequence in context: A019668 A091700 A135895 this_sequence A078301 A060583 A078104
%K A039814 sign,tabl,nice
%O A039814 1,2
%A A039814 Christian G. Bower (bowerc(AT)usa.net), Feb 15 1999.

%I A078301
%S A078301 2,1,7,6,7
%N A078301 Decimal expansion of Planck mass.
%H A078301 NIST Physics Laboratory, <a href="http://physics.nist.gov/cgi-bin/cuu/Value?plkm">Planck mass</a>
%H A078301 Eric Weisstein's World of Mathematics, World of Physics, <a href="http://scienceworld.wolfram.com/physics/PlanckMass.html">Planck Mass</a>
%H A078301 Wikipedia, <a href="http://en.wikipedia.org/wiki/Planck_mass">Planck mass</a>
%e A078301 Planck mass = 2.1767 * 10^-8 kilograms.
%Y A078301 Cf. A003676, A078300, A078302.
%Y A078301 Adjacent sequences: A078298 A078299 A078300 this_sequence A078302 A078303 A078304
%Y A078301 Sequence in context: A091700 A135895 A039814 this_sequence A060583 A078104 A072280
%K A078301 nonn,cons
%O A078301 -8,1
%A A078301 Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 21 2002

%I A060583
%S A060583 0,2,1,7,6,8,5,4,3,23,22,21,18,20,19,25,24,26,16,15,17,14,13,12,9,11,
%T A060583 10,70,69,71,68,67,66,63,65,64,54,56,55,61,60,62,59,58,57,77,76,75,72,
%U A060583 74,73,79,78,80,50,49,48,45,47,46,52,51,53,43,42,44,41,40,39,36,38,37
%N A060583 A ternary code related to the Tower of Hanoi.
%C A060583 Write n in base 3, then (working from left to right) if the k-th digit of n is equal to the corresponding digit to the left of the k-th digit of a(n) then this is the k-th digit of a(n), otherwise the k-th digit of a(n) is the element of {0,1,2} which has not just been compared, then read result as a base 3 number.
%H A060583 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A060583 a(n) =3a([n/3])+((-a([n/3])-n) mod 3) =3a([n/3])+A060582(n) with a(0)=0.
%e A060583 a(46)=76 since in base 3 43 is written as 1201; this gives a first digit of 2(=3-1-0) for a(n), a second digit of 2(=2=2), a third digit of 1(=3-2-0) and a fourth digit of 1(=1=1); 2211 base 3 is 76.
%Y A060583 Cf. A060586, A060587.
%Y A060583 Adjacent sequences: A060580 A060581 A060582 this_sequence A060584 A060585 A060586
%Y A060583 Sequence in context: A135895 A039814 A078301 this_sequence A078104 A072280 A086054
%K A060583 base,nonn
%O A060583 0,2
%A A060583 Henry Bottomley (se16(AT)btinternet.com), Apr 04 2001

%I A078104
%S A078104 1,0,2,1,7,6,37,42,237,320,1715,2610,13478,22404,112480,200158,
%T A078104 982561,1846314,8897089,17481864
%N A078104 Number of ways a loop can cross three roads meeting in a Y n times. The loop must touch the South-West sector.
%C A078104 The Mercedes-Benz problem: closed meanders crossing a Y.
%H A078104 Anonymous, <a href="http://www.research.att.com/~njas/sequences/a078105.a.gif">Illustration for a(3) = 1</a>
%e A078104 With three crossings the loop must cut each road exactly once, so a(3) = 1.
%e A078104 With 4 crossings the loop can cut one road 4 times (giving A005315(2)*2 = 4 possibilities), or two roads twice each (3 ways), so a(4) = 7.
%Y A078104 See A085919 for another version. Cf. A078105 (nonisomorphic solutions), A077460 and A005315 (loop crossing one road).
%Y A078104 Cf. also A077550.
%Y A078104 Adjacent sequences: A078101 A078102 A078103 this_sequence A078105 A078106 A078107
%Y A078104 Sequence in context: A039814 A078301 A060583 this_sequence A072280 A086054 A011134
%K A078104 nonn
%O A078104 0,3
%A A078104 njas and Jon Wild (wild(AT)music.mcgill.ca), Dec 05 2002
%E A078104 More terms added Aug 25, 2003

%I A072280
%S A072280 2,1,7,6,41,5,239,34,199,29,8119,33,47321,169,961,1154,1607521,197,
%T A072280 9369319,1121,32641,5741,318281039,1153
%N A072280 Sequence arising from factorization of the Pell numbers and the Companion Pell numbers f(n)=A000129 and L(n)=A002203. f(n)=2f(n-1)+f(n-2) L(n)=f(n-1)+f(n+1).
%C A072280 For even n, f(n)=Product(d|n)a(d); for odd n, f(n)=Product( d divides n )a(2d); for odd prime p, a(p)=L(p)/2, a(2p)=f(p) a(1)=2, a(2)=1; a(2^(k+1))=L(2^k); for odd n, L(n)=Product( d divides n )a(d); for k>0 and odd n, L(n*2^k)=Product( d divides n )a(d*2^(k+1))
%F A072280 h=1+sqrt(2). h^2=1+2h. 1/h=h-2. K(n, x)=n-th cyclotomic polynomial, so that x^n-1=Product( d divides n )K(d, x). g(d) is the order of K(d, x). a(n)=(h-2)^g(n)*K(n, h^2)
%e A072280 f(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12)=2*1*7*6*5*33=13860 for even n, f(9)=a(2)*a(6)*a(18)= 1*5*197=985 for odd n. L(12)=a(8)*a(24)=34*1153=39202 for even n, L(21)=a(1)*a(3)*a(7)*a(21)=2*7*239*32641=109216786.
%Y A072280 Cf. A000129, A002203.
%Y A072280 Adjacent sequences: A072277 A072278 A072279 this_sequence A072281 A072282 A072283
%Y A072280 Sequence in context: A078301 A060583 A078104 this_sequence A086054 A011134 A021463
%K A072280 nonn,uned
%O A072280 1,1
%A A072280 M. Kristof (kristmikl(AT)freemail.hu), Jul 10 2002

%I A086054
%S A086054 2,1,7,7,5,8,6,0,9,0,3,0,3,6,0,2,1,3,0,5,0,0,6,8,8,8,9,8,2,3,7,6,1,3,9,
%T A086054 4,7,3,3,8,5,8,3,7,0,0,3,6,9,2,8,6,2,9,4,3,2,5,7,9,5,2,5,3,1,9,4,3,0,8,
%U A086054 5,4,9,1,7,6,7,4,1,9,8,6,4,3,0,3,2,8,9,6,1,6,1,0,6,6,3,0,2,5,0,5,7,6,1
%N A086054 Decimal expansion of Pi*log(2).
%C A086054 Madelung constant b2(2), negated.
%H A086054 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>
%e A086054 2.1775860903036021305006888982376139...
%Y A086054 Adjacent sequences: A086051 A086052 A086053 this_sequence A086055 A086056 A086057
%Y A086054 Sequence in context: A060583 A078104 A072280 this_sequence A011134 A021463 A111479
%K A086054 nonn,cons,easy
%O A086054 1,1
%A A086054 Eric Weisstein (eric(AT)weisstein.com), Jul 07, 2003
%E A086054 Corrected by Antti Ahti (antti.ahti(AT)tkk.fi), Nov 17 2004
%E A086054 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), May 21 2005

%I A011134
%S A011134 2,1,7,7,9,0,6,4,2,4,4,8,2,7,7,9,8,9,4,6,6,5,2,6,4,8,3,5,5,7,5,2,0,
%T A011134 7,1,0,7,0,1,0,6,6,6,4,8,6,0,9,6,1,6,2,0,2,1,7,7,7,9,0,8,8,1,8,5,1,
%U A011134 9,1,4,9,3,6,7,1,3,6,9,2,4,3,5,4,5,2,0,6,2,4,3,7,9,7,5,4,5,5,4,3,7
%N A011134 Decimal expansion of 5th root of 49.
%Y A011134 Adjacent sequences: A011131 A011132 A011133 this_sequence A011135 A011136 A011137
%Y A011134 Sequence in context: A078104 A072280 A086054 this_sequence A021463 A111479 A102817
%K A011134 nonn,cons
%O A011134 1,1
%A A011134 njas

%I A021463
%S A021463 0,0,2,1,7,8,6,4,9,2,3,7,4,7,2,7,6,6,8,8,4,5,3,1,5,9,0,4,1,3,9,4,3,
%T A021463 3,5,5,1,1,9,8,2,5,7,0,8,0,6,1,0,0,2,1,7,8,6,4,9,2,3,7,4,7,2,7,6,6,
%U A021463 8,8,4,5,3,1,5,9,0,4,1,3,9,4,3,3,5,5,1,1,9,8,2,5,7,0,8,0,6,1,0,0,2
%N A021463 Decimal expansion of 1/459.
%Y A021463 Adjacent sequences: A021460 A021461 A021462 this_sequence A021464 A021465 A021466
%Y A021463 Sequence in context: A072280 A086054 A011134 this_sequence A111479 A102817 A026252
%K A021463 nonn,cons
%O A021463 0,3
%A A021463 njas

%I A111479
%S A111479 2,1,7,8,18
%N A111479 a(n) = least positive number such that (2 Concat Concat_{k=0}^n Repeat(A045572(n), a(n))) is a prime. A045572 = odd numbers not divisible by 5.
%C A111479 Concat is the operator that concatenates two numbers as decimal strings, and Repeat returns the number obtained by concatenating the first argument to itself the second argument number of times.
%e A111479 21 is not a prime, but 211 is, so a(0)=2.
%e A111479 211,2113,21137777777 are all primes.
%e A111479 21137777777 is a prime obtained as 2 followed by two 1's, one 3, and seven 7's, so a(2) = 7.
%Y A111479 Cf. A111471, A111472, A111473, A111474, A111475, A111477, A111478, A111480.
%Y A111479 Adjacent sequences: A111476 A111477 A111478 this_sequence A111480 A111481 A111482
%Y A111479 Sequence in context: A086054 A011134 A021463 this_sequence A102817 A026252 A032298
%K A111479 base,more,nonn,less
%O A111479 0,1
%A A111479 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2005
%E A111479 Edited by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 29 2006.

%I A102817
%S A102817 2,1,7,9,9,9,9,7,6,4,4,9,9,9,8,8,1,4,6,8,6,2,8,8,1,3,9,5,7,7,9,3,6,0,9,
%T A102817 8,9,0,7,2,6,7,9,7,8,9,0,9,7,3,0,0,5,6,5,4,8,3,2,8,8,5,2,1,2,2,4,0,4,2,
%U A102817 3,7,7,2,0,9,6,4,2,6,1,4,9,8,3,9,2,3,1,1,2,6,8,1,5,0,7,1,6,5,3,3,0,8,6
%N A102817 Decimal expansion of Gamma(delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).
%C A102817 Let x be this constant, then the Integral_{1...x} sin(t)/sqrt(t) dt = .655555692248871113068...
%C A102817 delta^2 = 21.8014436664499573..., (delta/Gamma(delta))^2 = .10000663312663433933000349...
%C A102817 If s is solution of Gamma(s) - sqrt(218) = 0 then 1/((s - delta)*Gamma(delta)^6) = 2.5555951358396... whereas a^(Pi/4) = 2.055596478435... where a is Feigenbaum alpha constant (A006891), the difference = 0.4999986574... ~ 1/(2 + 10^-5.27)
%C A102817 10*cos(Gamma(delta)^2) + Pi = -0.199999019922688714710053...
%H A102817 <a href="http://pi.lacim.uqam.ca/piDATA/feigenbaum.txt">Feigenbaum constants to 1018 decimal places</a>
%e A102817 217.99997644999881468628813957793609890726797890973...
%t A102817 Set delta then RealDigits[Gamma[delta]^2, 10, 110][[1]]
%Y A102817 Cf. A006890, A006891.
%Y A102817 Cf. A006891.
%Y A102817 Adjacent sequences: A102814 A102815 A102816 this_sequence A102818 A102819 A102820
%Y A102817 Sequence in context: A011134 A021463 A111479 this_sequence A026252 A032298 A032210
%K A102817 cons,nonn
%O A102817 3,1
%A A102817 Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Feb 26 2005

%I A026252
%S A026252 2,1,7,9,12,14,3,19,4,24,26,5,31,6,36,38,41,43,8,48,50,53,55,
%T A026252 10,60,11,65,67,70,72,13,77,79,82,84,15,89,16,94,96,17,101,18,
%U A026252 106,108,111,113,20,118,21,123,125,22,130,23,135,137,140,142
%N A026252 a(n) = (1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026250. Also a(n) = position of n in A026252.
%Y A026252 Adjacent sequences: A026249 A026250 A026251 this_sequence A026253 A026254 A026255
%Y A026252 Sequence in context: A021463 A111479 A102817 this_sequence A032298 A032210 A032135
%K A026252 nonn
%O A026252 1,1
%A A026252 Clark Kimberling (ck6(AT)evansville.edu)

%I A032298
%S A032298 2,1,7,9,21,148,281,997,3037,33036,76275,352586,1466713,6840499,
%T A032298 96591514,293606525,1501812845,7541619580,45272015807,
%U A032298 252241209910,5166271223047,18007882089737,113281855815782
%N A032298 "EFJ" (unordered, size, labeled) transform of 2,1,1,1,...
%H A032298 C. G. Bower, <a href="http://www.research.att.com/~njas/sequences/transforms2.html">Transforms (2)</a>
%Y A032298 Adjacent sequences: A032295 A032296 A032297 this_sequence A032299 A032300 A032301
%Y A032298 Sequence in context: A111479 A102817 A026252 this_sequence A032210 A032135 A032039
%K A032298 nonn
%O A032298 1,1
%A A032298 Christian G. Bower (bowerc(AT)usa.net)

%I A032210
%S A032210 2,1,7,9,21,148,281,997,3037,83436,187155,906986,4040713,19429075,
%T A032210 967161214,2687357885,15062567325,76117375036,495916450391,
%U A032210 2830109972230,258366446110567,832033559653913,5503200805013334
%N A032210 "DFJ" (bracelet, size, labeled) transform of 2,1,1,1...
%H A032210 C. G. Bower, <a href="http://www.research.att.com/~njas/sequences/transforms2.html">Transforms (2)</a>
%Y A032210 Adjacent sequences: A032207 A032208 A032209 this_sequence A032211 A032212 A032213
%Y A032210 Sequence in context: A102817 A026252 A032298 this_sequence A032135 A032039 A075118
%K A032210 nonn
%O A032210 1,1
%A A032210 Christian G. Bower (bowerc(AT)usa.net)

%I A032135
%S A032135 2,1,7,9,21,268,491,1893,5809,166476,373275,1812374,8077317,
%T A032135 38851660,1934306029,5374689421,30125069097,152234643292,
%U A032135 991832638619,5660219512530,516732891172537,1664067117563368
%N A032135 "CFJ" (necklace, size, labeled) transform of 2,1,1,1...
%H A032135 C. G. Bower, <a href="http://www.research.att.com/~njas/sequences/transforms2.html">Transforms (2)</a>
%H A032135 <a href="http://www.research.att.com/~njas/sequences/Sindx_Lu.html#Lyndon">Index entries for sequences related to Lyndon words</a>
%Y A032135 Adjacent sequences: A032132 A032133 A032134 this_sequence A032136 A032137 A032138
%Y A032135 Sequence in context: A026252 A032298 A032210 this_sequence A032039 A075118 A100245
%K A032135 nonn
%O A032135 1,1
%A A032135 Christian G. Bower (bowerc(AT)usa.net)

%I A032039
%S A032039 2,1,7,9,21,388,701,2789,8581,325116,725715,3549362,15974921,
%T A032039 77157109,4774872394,13166159997,74350315389,375874959532,
%U A032039 2464017881123,14085616954310,1537975786987987,4933351699929687
%N A032039 "BFJ" (reversible, size, labeled) transform of 2,1,1,1...
%H A032039 C. G. Bower, <a href="http://www.research.att.com/~njas/sequences/transforms2.html">Transforms (2)</a>
%Y A032039 Adjacent sequences: A032036 A032037 A032038 this_sequence A032040 A032041 A032042
%Y A032039 Sequence in context: A032298 A032210 A032135 this_sequence A075118 A100245 A095137
%K A032039 nonn
%O A032039 1,1
%A A032039 Christian G. Bower (bowerc(AT)usa.net)

%I A075118
%S A075118 2,1,7,10,31,61,154,337,799,1810,4207,9637,22258,51169,117943,271450,
%T A075118 625279,1439629,3315466,7634353,17580751,40483810,93226063,214677493,
%U A075118 494355682,1138388161,2621455207,6036619690,13900985311,32010844381
%N A075118 Variant on Lucas numbers: a(n)=a(n-1)+3*a(n-2) with a(0)=2 and a(1)=1.
%C A075118 The sequence 4,1,7,.. = 2*0^n+A075118(n) is given by trace(A^n) where A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry (pbarry(AT)wit.ie), Oct 01 2004
%D A075118 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.
%F A075118 a(n) = ((1+sqrt(13))/2)^n+((1-sqrt(13))/2)^n = 2*A006130(n)-A006130(n-1) = A075117(3, n).
%e A075118 a(4) = a(3)+3*a(2) = 10+3*7 = 31.
%t A075118 a[0] = 2; a[1] = 1; a[n_] := a[n] = a[n - 1] + 3a[n - 2]; Table[ a[n], {n, 0, 30}]
%Y A075118 Cf. A000032, A006130, A014551, A072265, A075117.
%Y A075118 Adjacent sequences: A075115 A075116 A075117 this_sequence A075119 A075120 A075121
%Y A075118 Sequence in context: A032210 A032135 A032039 this_sequence A100245 A095137 A113042
%K A075118 easy,nonn
%O A075118 0,1
%A A075118 Henry Bottomley (se16(AT)btinternet.com), Sep 02 2002

%I A100245
%S A100245 1,1,2,1,7,11,3,1,12,44,56,18,1,17,102,267,302,123,11,1,22,185,758,1597,
%T A100245 1670,757,106,1,27,293,1654,5256,9503,9401,4603,908,41,1,32,426,3080,
%U A100245 13254,35004,56456,53588,27688,6716,540,1,37,584,5161,28191,99183
%N A100245 Triangle read by rows: T(n,k) is the number of k-matchings in the P_3 X P_n lattice graph.
%C A100245 Row n contains 1+floor(3n/2) terms. Row sums yield A033506.
%D A100245 H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (26) and Table V).
%F A100245 G.f.=(1+tz-t^3*z^2)(1-2tz-t^3*z^2)/[1-(1+3t)z-t(1+t)(2+5t)z^2-t^2*(1+2t)(1-t)z^3+t^4*(2+3t+5t^2)z^4-t^6*(1-t)z^5-t^9*z^6]. The row generating polynomials A[n] satisfy A[n]=(1+3t)A[n-1]+t(2+7t+5t^2)A[n-2]+t^2*(1+t-2t^2)A[n-3]-t^4*(2+3t+5t^2)A[n-4]+t^6*(1-t)A[n-5]+t^9*A[n-6].
%e A100245 T(2,2)=11 because in the P_3 X P_ 2 lattice graph with vertex set {O(0,0),A(1,0),B(1,1),C(1,2),D(0,2),E(0,1)} and edge set {OA,EB,DC,OE,ED,AB,BC} we have the following eleven 2-matchings: {OA,EB},{OA,DC},{EB,DC},{OA,ED},{OA,BC},{DC,OE},{DC,AB},{OE,AB},{OE,BC},{ED,AB}, and {ED,BC}.
%e A100245 Triangle starts:
%e A100245 1;
%e A100245 1,2;
%e A100245 1,7,11,3;
%e A100245 1,12,44,56,18;
%e A100245 1,17,102,267,302,123,11;
%p A100245 G:=(1+t*z-t^3*z^2)*(1-2*t*z-t^3*z^2)/(1-(1+3*t)*z-t*(1+t)*(2+5*t)*z^2-t^2*(1+2*t)*(1-t)*z^3+t^4*(2+3*t+5*t^2)*z^4-t^6*(1-t)*z^5-t^9*z^6): Gser:=simplify(series(G,z=0,11)): P[0]:=1: for n from 1 to 8 do P[n]:=coeff(Gser,z^n) od:for n from 0 to 8 do seq(coeff(t*P[n],t^k),k=1..floor(3*n/2)+1) od; # yields sequence in triangular form
%Y A100245 Cf. A033506, A001835.
%Y A100245 Adjacent sequences: A100242 A100243 A100244 this_sequence A100246 A100247 A100248
%Y A100245 Sequence in context: A032135 A032039 A075118 this_sequence A095137 A113042 A013070
%K A100245 nonn,tabl
%O A100245 0,3
%A A100245 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2004

%I A095137
%S A095137 2,1,7,11,89,163,1597,3317,37823,107413,1182887,4232341,49100059,
%T A095137 184657283,2329965377,10114830259,138903895201,622143222539,
%U A095137 9382665690241,44778520855589,686482057860331,3598441529151191
%N A095137 Absolute difference between the product of the first [n/2] even indexed primes and the product of the first [n/2] odd indexed primes.
%F A095137 the absolute difference of the product_{j=2..n, 2} p_j (A066206) and the product_{k=1..n, 2} p_j (A066205).
%e A095137 a(5) = 2*5*11 - 3*7 = 89, a(6) = 3*7*13 - 2*5*11 = 163, a(7) =
%e A095137 2*5*11*17 - 3*7*13 = 1597, a(8) = 3*7*13*19 - 2*5*11*17 = 3317.
%t A095137 PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[n]]; f[n_] := Abs[ Product[ Prime[i], {i, 2, n, 2}] + Product[ Prime[i], {i, 1, n, 2}]]; f[1] = 2; Table[ f[n], {n, 24}]
%Y A095137 Cf. A095134, A000040, A066206, A066205.
%Y A095137 Adjacent sequences: A095134 A095135 A095136 this_sequence A095138 A095139 A095140
%Y A095137 Sequence in context: A032039 A075118 A100245 this_sequence A113042 A013070 A012888
%K A095137 nonn
%O A095137 1,1
%A A095137 Robert G. Wilson v (rgwv(AT)rgwv.com), May 28 2004

%I A113042
%S A113042 0,2,1,7,15,45,139,438,1419,4703,16019,55146,190254,671215,2404179,
%T A113042 8534995,30635448,110495549,401418693,1467388464,5393131894,19883104535,
%U A113042 73856058401,273600682457,1017557492609,3803885439979,14266466901249
%N A113042 Number of solutions to +-p(1)+-p(2)+-...+-p(2n)=3 where p(i) is the i-th prime.
%C A113042 +-p(1)+-p(2)+-...+-p(2n+1)=3 does not have solutions.
%p A113042 A113042:=proc(n) local i,j,p,t; t:= NULL; for j from 2 to 2*n by 2 do p:=1; for i to j do p:=p*(x^(-ithprime(i))+x^(ithprime(i))); od; t:=t,coeff(p,x,3); od; t; end;
%Y A113042 Cf. A022894.
%Y A113042 Adjacent sequences: A113039 A113040 A113041 this_sequence A113043 A113044 A113045
%Y A113042 Sequence in context: A075118 A100245 A095137 this_sequence A013070 A012888 A012893
%K A113042 nonn
%O A113042 1,2
%A A113042 Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 12 2005

%I A013070
%S A013070 2,1,7,18,5,135,2510,13300,22895,456075,10766650,86950050,224989375,
%T A013070 6482478275,213120778000,2265017781000,5257017140375,268111942824375,
%U A013070 12649162007056250,161640657722878750,19598816145861875
%V A013070 2,-1,-7,18,-5,135,-2510,13300,-22895,456075,-10766650,86950050,
%W A013070 -224989375,6482478275,-213120778000,2265017781000,-5257017140375,
%X A013070 268111942824375,-12649162007056250,161640657722878750
%N A013070 sin(arcsinh(x)+log(x+1))=2*x-1/2!*x^2-7/3!*x^3+18/4!*x^4-5/5!*x^5...
%Y A013070 Adjacent sequences: A013067 A013068 A013069 this_sequence A013071 A013072 A013073
%Y A013070 Sequence in context: A100245 A095137 A113042 this_sequence A012888 A012893 A013075
%K A013070 sign
%O A013070 0,1
%A A013070 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)

%I A012888
%S A012888 2,1,7,18,13,135,1614,10612,71215,637515,6443034,67830114,769960735,
%T A012888 9523596355,126475570464,1788373677960,26860585215767,
%U A012888 427142221117623,7163185240710714,126244734481260190
%V A012888 2,-1,-7,18,-13,135,-1614,10612,-71215,637515,-6443034,67830114,
%W A012888 -769960735,9523596355,-126475570464,1788373677960,-26860585215767,
%X A012888 427142221117623,-7163185240710714,126244734481260190
%N A012888 sin(sin(x)+log(x+1))=2*x-1/2!*x^2-7/3!*x^3+18/4!*x^4-13/5!*x^5...
%Y A012888 Adjacent sequences: A012885 A012886 A012887 this_sequence A012889 A012890 A012891
%Y A012888 Sequence in context: A095137 A113042 A013070 this_sequence A012893 A013075 A009281
%K A012888 sign
%O A012888 0,1
%A A012888 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)

%I A012893
%S A012893 2,1,7,18,243,1785,19086,308980,2736225,90239805,495769586,
%T A012893 38522052690,39766893593,22481798964989,111035038388104,
%U A012893 16941789968380680,219516343596465735,15750892172912678361
%V A012893 2,-1,-7,18,243,-1785,-19086,308980,2736225,-90239805,-495769586,
%W A012893 38522052690,39766893593,-22481798964989,111035038388104,
%X A012893 16941789968380680,-219516343596465735,-15750892172912678361
%N A012893 arcsinh(sin(x)+log(x+1))=2*x-1/2!*x^2-7/3!*x^3+18/4!*x^4+243/5!*x^5...
%Y A012893 Adjacent sequences: A012890 A012891 A012892 this_sequence A012894 A012895 A012896
%Y A012893 Sequence in context: A113042 A013070 A012888 this_sequence A013075 A009281 A085073
%K A012893 sign
%O A012893 0,1
%A A012893 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)

%I A013075
%S A013075 2,1,7,18,251,1785,19982,311668,2913569,91711485,552487698,
%T A013075 39470080914,65100175929,23245363302525,96919211293848,
%U A013075 17705730882051720,211105141092307143,16681025379015259353
%V A013075 2,-1,-7,18,251,-1785,-19982,311668,2913569,-91711485,-552487698,
%W A013075 39470080914,65100175929,-23245363302525,96919211293848,
%X A013075 17705730882051720,-211105141092307143,-16681025379015259353
%N A013075 arcsinh(arcsinh(x)+log(x+1))=2*x-1/2!*x^2-7/3!*x^3+18/4!*x^4+251/5!*x^5...
%Y A013075 Adjacent sequences: A013072 A013073 A013074 this_sequence A013076 A013077 A013078
%Y A013075 Sequence in context: A013070 A012888 A012893 this_sequence A009281 A085073 A131288
%K A013075 sign
%O A013075 0,1
%A A013075 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)

%I A009281
%S A009281 1,1,2,1,7,19,136,923,7421,66743,667486,7342279,88107427,1145396459,
%T A009281 16035550532,240533257859,3848532125881,65425046139823,
%U A009281 1177650830516986,22375365779822543,447507315596451071
%V A009281 1,1,2,1,7,-19,136,-923,7421,-66743,667486,-7342279,88107427,-1145396459,
%W A009281 16035550532,-240533257859,3848532125881,-65425046139823,
%X A009281 1177650830516986,-22375365779822543,447507315596451071
%N A009281 Expansion of exp(x)*cosh(ln(1+x)).
%t A009281 Exp[ x ]*Cosh[ Log[ 1+x ] ]
%Y A009281 Adjacent sequences: A009278 A009279 A009280 this_sequence A009282 A009283 A009284
%Y A009281 Sequence in context: A012888 A012893 A013075 this_sequence A085073 A131288 A111789
%K A009281 sign,easy
%O A009281 0,3
%A A009281 R. H. Hardin (rhh(AT)cadence.com)
%E A009281 Extended with signs Mar 15 1997 by Olivier Gerard.

%I A085073
%S A085073 2,1,7,41,15,134,3,127,11,2,3,548,2,1,3,389,5,582,2,316,1,38,3,2216,3,2,
%T A085073 13,212,5,2742,2,1669,1,1,31,2764,2,1,13,1094,4,2298,3,1,123,14,11,8912,
%U A085073 3,202,17,2,2,1146,23,904,1,26,3,11028,13,22,57,3581,37,1194,2,172,15
%N A085073 Smallest k such that n+k and nk have the same prime signature, or 0 if no such number exists.
%e A085073 a(6) = 379 as 6*379 = 2*3*379 and 6+379 = 385 = 5*7*11 both have prime signature p*q*r.
%Y A085073 Cf. A085072.
%Y A085073 Adjacent sequences: A085070 A085071 A085072 this_sequence A085074 A085075 A085076
%Y A085073 Sequence in context: A012893 A013075 A009281 this_sequence A131288 A111789 A021825
%K A085073 nonn
%O A085073 1,1
%A A085073 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 01 2003
%E A085073 Corrected by Jason Earls (jcearls(AT)cableone.net), Jul 10 2003
%E A085073 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jan 12 2005

%I A131288
%S A131288 2,1,7,193,63775,4294321153,18446744022173838463,
%T A131288 340282366920938463205120190760593525761,
%U A131288 115792089237316195423570985008687907847825466794905548626109625623336235655679
%N A131288 Number of sets of subsets of an n-set such that the union is the whole set and the intersection is empty (not excluding the whole or the empty from being in the cover).
%C A131288 Number of covering sets of subsets of n-set is inverse binomial transform of number of sets of subsets. Number of coverings with empty intersection is (to within a unit parity flutter and a fudge unit when n = 0) inverse binomial transform of number of coverings, i.e. second inverse binomial transform of number of sets of subsets.
%F A131288 0^n - (-1)^n + double sum on k from 0 to n and t from 0 to k: (n choose k) (k choose t) (-1)^(n-t) 2^(2^t)
%Y A131288 Cf. A003465 (coverings by non-empty subsets), A000371 = 2 x A003465 (coverings allowing empty block).
%Y A131288 Adjacent sequences: A131285 A131286 A131287 this_sequence A131289 A131290 A131291
%Y A131288 Sequence in context: A013075 A009281 A085073 this_sequence A111789 A021825 A011327
%K A131288 nonn,nice
%O A131288 0,1
%A A131288 David Pasino (davepasino(AT)yahoo.com), Sep 29 2007

%I A111789
%S A111789 2,1,7,120539
%N A111789 First differences of [0, A111788], the sequence that begins with 0 and continues with the terms of A111788.
%Y A111789 Cf. A111788, A111790.
%Y A111789 Adjacent sequences: A111786 A111787 A111788 this_sequence A111790 A111791 A111792
%Y A111789 Sequence in context: A009281 A085073 A131288 this_sequence A021825 A011327 A083529
%K A111789 nonn
%O A111789 0,1
%A A111789 Jon Awbrey (jawbrey(AT)att.net), Aug 17 2005

%I A021825
%S A021825 0,0,1,2,1,8,0,2,6,7,9,6,5,8,9,5,2,4,9,6,9,5,4,9,3,3,0,0,8,5,2,6,1,
%T A021825 8,7,5,7,6,1,2,6,6,7,4,7,8,6,8,4,5,3,1,0,5,9,6,8,3,3,1,3,0,3,2,8,8,
%U A021825 6,7,2,3,5,0,7,9,1,7,1,7,4,1,7,7,8,3,1,9,1,2,3,0,2,0,7,0,6,4,5,5,5
%N A021825 Decimal expansion of 1/821.
%Y A021825 Adjacent sequences: A021822 A021823 A021824 this_sequence A021826 A021827 A021828
%Y A021825 Sequence in context: A085073 A131288 A111789 this_sequence A011327 A083529 A102208
%K A021825 nonn,cons
%O A021825 0,4
%A A021825 njas

%I A011327
%S A011327 1,2,1,8,1,1,4,0,4,3,5,6,0,7,6,8,1,9,6,6,5,9,0,2,5,7,3,3,3,7,3,6,6,
%T A011327 7,1,5,9,3,2,8,0,7,9,8,5,3,1,9,9,7,2,0,9,7,5,8,7,2,6,5,9,8,7,9,2,1,
%U A011327 0,6,1,4,2,4,5,3,2,0,7,6,9,4,6,9,9,2,2,3,3,5,3,0,3,8,4,7,2,8,6,7,7
%N A011327 Decimal expansion of 13th root of 13.
%Y A011327 Adjacent sequences: A011324 A011325 A011326 this_sequence A011328 A011329 A011330
%Y A011327 Sequence in context: A131288 A111789 A021825 this_sequence A083529 A102208 A009385
%K A011327 nonn,cons
%O A011327 1,2
%A A011327 njas

%I A083529
%S A083529 2,1,8,1,5,1,5,1,26,25,5,1,5,25,35,1,5,1,5,25,62,25,5,1,50,25,80,37,5,
%T A083529 55,5,1,26,25,80,1,5,25,8,25,5,1,5,97,80,25,5,1,68,25,125,1,5,1,155,25,
%U A083529 125,25,5,145,5,25,188,1,5,181,5,13,125,205,5,1,5,25,125,169,80,181,5
%N A083529 5^n mod 3n.
%e A083529 n=3: modulus=3n=9; 5^n=5^3=125=9.13+8=9.13+a(3).
%t A083529 Table[Mod[5^w, 3*w], {w, 1, 100}]
%Y A083529 Cf. A000079, A000244, A000351, A000420, A082511.
%Y A083529 Cf. A083528, A083530.
%Y A083529 Adjacent sequences: A083526 A083527 A083528 this_sequence A083530 A083531 A083532
%Y A083529 Sequence in context: A111789 A021825 A011327 this_sequence A102208 A009385 A008308
%K A083529 easy,nonn
%O A083529 1,1
%A A083529 Labos E. (labos(AT)ana.sote.hu), Apr 30 2003

%I A102208
%S A102208 2,1,8,1,6,9,4,9,9,0,6,2,4,9,1,2,3,7,3,5,0,3,8,2,2,3,6,1,9,7,1,3,6,5,0,
%T A102208 9,8,1,0,0,2,5,7,6,4,9,8,3,8,1,3,5,7,1,8,4,4,6,2,0,7,1,8,5,5,8,7,7,1,7,
%U A102208 0,5,2,3,4,9,0,8,5,3,7,4,7,5,6,0,0,6,0,0,3,4,9,1,1,5,9,2,8,1
%N A102208 Decimal expansion of volume of an icosahedron with each edge of unit length.
%H A102208 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Icosahedron.html">Icosahedron</a>
%F A102208 5/12 * (3 + sqrt(5))
%Y A102208 Adjacent sequences: A102205 A102206 A102207 this_sequence A102209 A102210 A102211
%Y A102208 Sequence in context: A021825 A011327 A083529 this_sequence A009385 A008308 A118931
%K A102208 cons,nonn
%O A102208 1,1
%A A102208 Bryan Jacobs (bryanjj(AT)gmail.com), Feb 17 2005

%I A009385
%S A009385 0,1,1,1,2,1,8,1,16,1,2304,1,68352,1,843776,1,221382656,1,9865887744,
%T A009385 1,888186732544,1,200433787207680,1,8060855544971264,1,
%U A009385 3814697273321848832,1,890946236332166348800,1,9668703713250158575616
%V A009385 0,1,-1,1,-2,1,8,1,-16,1,-2304,1,68352,1,843776,1,-221382656,1,9865887744,
%W A009385 1,888186732544,1,-200433787207680,1,8060855544971264,1,
%X A009385 3814697273321848832,1,-890946236332166348800,1,-9668703713250158575616
%N A009385 Expansion of ln(1+tanh(sinh(x))).
%t A009385 Log[ 1+Tanh[ Sinh[ x ] ] ]
%Y A009385 Adjacent sequences: A009382 A009383 A009384 this_sequence A009386 A009387 A009388
%Y A009385 Sequence in context: A011327 A083529 A102208 this_sequence A008308 A118931 A101280
%K A009385 sign,easy
%O A009385 0,5
%A A009385 R. H. Hardin (rhh(AT)cadence.com)
%E A009385 Extended with signs Mar 15 1997 by Olivier Gerard.

%I A008308
%S A008308 1,1,2,1,8,1,16,20,1,136,40,1,272,616,70,1,3968,2016,112,1,7936,28160,
%T A008308 5376,168,1,176896,135680,12432,240,1,353792,1805056,508640,25872,330,
%U A008308 1,11184128,11977856,1595264,49632,440,1,22368256,154918400,59835776
%N A008308 Triangle of tangent numbers.
%D A008308 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
%e A008308 1; 1; 2,1; 8,1; 16,20,1; 136,40,1; ...
%Y A008308 Essentially the same triangle as A059419, which is the main entry for this triangle.
%Y A008308 Adjacent sequences: A008305 A008306 A008307 this_sequence A008309 A008310 A008311
%Y A008308 Sequence in context: A083529 A102208 A009385 this_sequence A118931 A101280 A008309
%K A008308 tabf,nonn,nice
%O A008308 1,3
%A A008308 njas
%E A008308 More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

%I A118931
%S A118931 1,1,1,1,2,1,8,1,20,1,40,40,1,70,280,1,112,1120,1,168,3360,2240,1,240,
%T A118931 8400,22400,1,330,18480,123200,1,440,36960,492800,246400,1,572,68640,
%U A118931 1601600,3203200,1,728,120120,4484480,22422400,1,910,200200,11211200
%N A118931 Triangle, read by rows, where T(n,k) = n!/[k!*(n-3*k)!*3^k)] for n>=3*k>=0.
%C A118931 Row n contains 1+floor(n/3) terms. Row sums yield A001470. Given column vector V = A118932, then V is invariant under matrix product T*V = V, or, A118932(n) = Sum_{k=0..n} T(n,k)*A118932(k). Given C = Pascal's triangle and T = this triangle, then matrix product M = C^-1*T yields M(3n,n) = (3*n)!/(n!*3^n), 0 otherwise (cf. A100861 formula due to Paul Barry).
%F A118931 E.g.f.: A(x,y) = exp(x + y*x^3/3).
%e A118931 Triangle T begins:
%e A118931 1;
%e A118931 1;
%e A118931 1;
%e A118931 1,2;
%e A118931 1,8;
%e A118931 1,20;
%e A118931 1,40,40;
%e A118931 1,70,280;
%e A118931 1,112,1120;
%e A118931 1,168,3360,2240;
%e A118931 1,240,8400,22400;
%e A118931 1,330,18480,123200;
%e A118931 1,440,36960,492800,246400; ...
%o A118931 (PARI) T(n,k)=if(n<3*k,0,n!/(k!*(n-3*k)!*3^k))
%Y A118931 Cf. A001470 (row sums), A118932 (invariant vector); variants: A100861, A118933.
%Y A118931 Adjacent sequences: A118928 A118929 A118930 this_sequence A118932 A118933 A118934
%Y A118931 Sequence in context: A102208 A009385 A008308 this_sequence A101280 A008309 A131175
%K A118931 nonn,tabl
%O A118931 0,5
%A A118931 Paul D. Hanna (pauldhanna(AT)juno.com), May 06 2006

%I A101280
%S A101280 1,1,1,2,1,8,1,22,16,1,52,136,1,114,720,272,1,240,3072,3968,1,494,11616,
%T A101280 34304,7936,1,1004,40776,230144,176896,1,2026,136384,1328336,2265344,
%U A101280 353792,1,4072,441568,6949952,21953408,11184128,1,8166,1398000,33981760
%N A101280 Triangle T(n,k) (n >= 1, 0 <= k <= [(n-1)/2]) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).
%C A101280 Row n has ceil(n/2) terms.
%C A101280 The paper by Shapiro et al. explains why T(n,k) is the number of permutations of n elements having k peaks and with the further property that every rise (ascent) is immediately followed by a peak. [That is, the permutation p_1 ... p_n has the further property that (j>1 and p_{j-1}<p_j) implies (j<n and p_j>p_{j+1}).] For example, the T(4,1)=8 permutations in the case n=4, k=1 are 1423, 2143, 2431, 3142, 3241, 3421, 4231, 4132.
%C A101280 A more elegant way to state this property: T(n,k) is the number of permutations of n objects with k descents such that every descent is a peak. The eight examples for n=4 and k=1 are now 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
%D A101280 L. Carlitz and Richard Scoville, Generalized Eulerian numbers: combinatorial applications, Journal f\"ur die reine und angewandte Mathematik 265 (1974), 111.
%D A101280 D. Foata and M. P. Sch\"utzenberger, Th\'eorie g\'eometrique des polyn\^omes eul\'eriens, Lecture Notes in Math. 138 (1970), 81-83.
%D A101280 D. Foata and V. Strehl, "Euler numbers and variations of permutations", in Colloquio Internazionale sulle Teorie Combinatorie, Rome, September 1973, (Atti dei Convegni Lincei 17, Rome, 1976), 129.
%D A101280 Louis W. Shapiro, Wen-Jin Woan and Seyoum Getu, "Runs, slides and moments", SIAM J. Algebraic and Discrete Methods 4 (1983), 461.
%F A101280 G.f.: sum_{n=1..infty, k=0..infty} T(n, k) t^k z^n/n! = C(t)(2-C(t))/(exp^{-z sqrt{1-4t}}+1-C(t))-C(t), where the sum on k is actually finite, running up to ceiling(n/2) - 1; here C(t) = (1-\sqrt{1-4t})/2t is the generating function for the Catalan numbers (A000108).
%F A101280 Sum_{k} Eulerian(n, k) x^k = sum_{k} T(n, k) x^k (1+x)^{n-1-2k}. E.g. 1+11x+11x^2+x^3 = (1+x)^3 + 8x(1+x).
%e A101280 Triangle begins:
%e A101280 1
%e A101280 1
%e A101280 1 2
%e A101280 1 8
%e A101280 1 22 16
%e A101280 1 52 136
%e A101280 1 114 720 272
%e A101280 ...
%p A101280 T:=proc(n,k) if k<0 then 0 elif n=1 and k=0 then 1 elif k>floor((n-1)/2) then 0 else (k+1)*T(n-1,k)+(2*n-4*k)*T(n-1,k-1) fi end: for n from 1 to 13 do seq(T(n,k),k=0..floor((n-1)/2)) od; # yields sequence in triangular form (Deutsch)
%Y A101280 The numbers 2^{n-1-k} T(n,k) form the array shown in A008303.
%Y A101280 Adjacent sequences: A101277 A101278 A101279 this_sequence A101281 A101282 A101283
%Y A101280 Sequence in context: A009385 A008308 A118931 this_sequence A008309 A131175 A066532
%K A101280 nonn,tabf,easy
%O A101280 1,4
%A A101280 D. E. Knuth, Jan 28 2005
%E A101280 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 06 2005

%I A008309
%S A008309 1,1,2,1,8,1,24,20,1,184,40,1,720,784,70,1,8448,2464,112,1,
%T A008309 40320,52352,6384,168,1,648576,229760,14448,240,1,3628800,
%U A008309 5360256,804320,29568,330,1
%V A008309 1,1,-2,1,-8,1,24,-20,1,184,-40,1,-720,784,-70,1,-8448,2464,-112,1,
%W A008309 40320,-52352,6384,-168,1,648576,-229760,14448,-240,1,-3628800,
%X A008309 5360256,-804320,29568,-330,1
%N A008309 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
%D A008309 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
%F A008309 E.g.f.: arctan(x)^k/k!=sum {n=0..inf} T(m, [ k+1 ]/2) x^m/m! where m=2n+k%2.
%e A008309 1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-200,0,1; 0,184,0,-40,0,1; ...
%o A008309 (PARI) T(n,k)=polcoeff(serlaplace(a(2*k-n%2)), n) where a(n)=atan(x)^n/n!
%Y A008309 Essentially same as A049218.
%Y A008309 A007290(n)=-T(n, [ (n-1)/2 ]), A010050(n)=(-1)^n*T(2n+1, 1), A049034(n)=(-1)^n*T(2n+2, 1), A049214(n)=(-1)^n*T(2n+3, 2), A049215(n)=(-1)^n*T(2n+4, 2), A049216(n)=(-1)^n*T(2n+5, 3), A049217(n)=(-1)^n*T(2n+6, 3).
%Y A008309 Adjacent sequences: A008306 A008307 A008308 this_sequence A008310 A008311 A008312
%Y A008309 Sequence in context: A008308 A118931 A101280 this_sequence A131175 A066532 A020778
%K A008309 sign,tabf,nice
%O A008309 1,3
%A A008309 njas
%E A008309 Additional comments from Michael Somos.

%I A131175
%S A131175 1,2,1,8,1,26,4,1,66,36,1,174,196,1,398,676,1,878,3044,1,2174,6852,192,
%T A131175 1,4862,18628,704,1,10494,45508,1216,1,22014,141252,6336,1,47614,315332,
%U A131175 10432,1,100862,858052,55488,1,225278,1878980,245952
%V A131175 1,-2,1,-8,1,-26,-4,1,-66,-36,1,-174,-196,1,-398,-676,1,-878,-3044,1,-2174,-6852,-192,
%W A131175 1,-4862,-18628,-704,1,-10494,-45508,-1216,1,-22014,-141252,-6336,1,-47614,-315332,
%X A131175 -10432,1,-100862,-858052,-55488,1,-225278,-1878980,-245952
%N A131175 Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.
%C A131175 Because the first column of A is a column vector of powers of 2, the determinant (for n>1) is always 0. Hence the rank is always (for n>1) less than n. A[n.n] = n-th n-almost prime A101695. The second column of the table is the negative of the trace of the matrices.
%F A131175 Row n of the table consists of the coefficients of x^n, x^n-1, ... of the characteristic polynomial of the n X n matrix A whose 1st row is the first n primes (1-almost primes) (A000040), 2nd row is the 1st n semiprimes (2-almost primes) A001358, 3rd row is the 1st n 3-almost primes A014612.
%e A131175 A_1 = [2], with determinant = 2 and characteristoic polynomial = x-2, with coefficients (1, -2) so a(a) = 1 and a(2) = -2.
%e A131175 A_2 =
%e A131175 [2.3]
%e A131175 [4.6]
%e A131175 with determinant = 0, polynomial x^2 - 8x, so the coefficients are (1, -8), hence a(3) = 1 and a(4) = -8.
%e A131175 A_3 =
%e A131175 [2..3..5]
%e A131175 [4..6..9]
%e A131175 [8.12.18]
%e A131175 with determinant = 0, polynomial = x^3 - 26x^2, -4x, so coefficients are (1, -26, -4), hence a(5) = 1, a(6) = -26, a(7) = -4.
%p A131175 A078840 := proc(n,m) local p,k ; k := 1 ; p := 2^n ; while k < m do p := p+1 ; while numtheory[bigomega](p) <> n do p := p+1 ; od; k := k+1 ; od: RETURN(p) ; end: A131175 := proc(nrow,showall) local A,row,col,pol,T,a ; A := linalg[matrix](nrow,nrow) ; for row from 1 to nrow do for col from 1 to nrow do if row = col then A[row,col] := x-A078840(row,col) ; else A[row,col] := -A078840(row,col) ; fi ; od: od: pol := linalg[det](A) ; T := [] ; for col from nrow to 0 by -1 do a := coeftayl(pol,x=0,col) ; if a <> 0 or showall then T := [op(T),a] ; fi ; od; RETURN(T) ; end: for n from 1 to 15 do print(op(A131175(n,false))) ; od: - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2007
%Y A131175 Cf. A000040, A001358, A014612, A014613, A014614, A101695.
%Y A131175 Adjacent sequences: A131172 A131173 A131174 this_sequence A131176 A131177 A131178
%Y A131175 Sequence in context: A118931 A101280 A008309 this_sequence A066532 A020778 A118961
%K A131175 sign
%O A131175 1,2
%A A131175 Jonathan Vos Post (jvospost2(AT)yahoo.com), Sep 24 2007
%E A131175 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2007

%I A066532
%S A066532 1,2,1,8,1,32,1,128,1,512,1,2048,1,8192,1,32768,1,131072,1,524288,1,
%T A066532 2097152,1,8388608,1,33554432,1,134217728,1,536870912,1,2147483648,1,
%U A066532 8589934592,1,34359738368,1,137438953472,1,549755813888,1
%N A066532 If n is odd a(n) = 1, if n is even a(n) = 2^(n-1).
%C A066532 Size of Frattini subgroup of the group of n X n signed permutations matrices (described in sequence A066051).
%F A066532 G.f.: G.f.: 1/(1-x^2)+2x(1+2x^2)/(1-2x^2); a(n)=2^n*(1-(-1)^n)/2+(1+(-1)^n)/2. - Paul Barry (pbarry(AT)wit.ie), Jun 17 2006
%t A066532 Table[ If[ OddQ[n], 1, 2^(n - 1)], {n, 1, 42} ]
%Y A066532 Cf. A066051.
%Y A066532 Adjacent sequences: A066529 A066530 A066531 this_sequence A066533 A066534 A066535
%Y A066532 Sequence in context: A101280 A008309 A131175 this_sequence A020778 A118961 A114706
%K A066532 nonn,easy
%O A066532 1,2
%A A066532 Sharon Sela (sharonsela(AT)hotmail.com), Jan 06 2002
%E A066532 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 07 2002
%E A066532 More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 25 2003

%I A020778
%S A020778 2,1,8,2,1,7,8,9,0,2,3,5,9,9,2,3,8,1,2,6,6,0,9,7,4,8,5,4,1,5,6,1,9,
%T A020778 4,5,1,8,5,6,4,0,2,6,9,4,1,3,1,8,0,8,1,8,5,8,3,8,4,4,0,1,0,1,1,3,8,
%U A020778 4,2,2,3,0,5,9,7,8,4,6,5,2,8,0,3,1,4,4,9,3,4,0,7,5,9,4,9,2,5,9,5,3
%N A020778 Decimal expansion of 1/sqrt(21).
%Y A020778 Adjacent sequences: A020775 A020776 A020777 this_sequence A020779 A020780 A020781
%Y A020778 Sequence in context: A008309 A131175 A066532 this_sequence A118961 A114706 A046740
%K A020778 nonn,cons
%O A020778 0,1
%A A020778 njas

%I A118961
%S A118961 1,1,2,1,8,2,1,8,1,9,2,18,8,1,9,25,2,18,1,8,32,2,1,9,25,49,8,1,32,9,25,
%T A118961 2,49,18,50,1,8,25,32,2,18,1,9,25,50,49,8,81,32,1,9,72,2,49,50,81,8,1,
%U A118961 98,32,25,49,72,2,18,1,121,9,25,49,8,98,32,81,121,1,9,2,25,18,128,49,50
%N A118961 Difference between long leg and hypotenuse in primitive Pythagorean triangles,sorted on hypotenuse(A020882), then on long leg(A046087).
%C A118961 Entries take only values appearing in A096033.
%Y A118961 Cf. A118962.
%Y A118961 Adjacent sequences: A118958 A118959 A118960 this_sequence A118962 A118963 A118964
%Y A118961 Sequence in context: A131175 A066532 A020778 this_sequence A114706 A046740 A130562
%K A118961 nonn
%O A118961 1,3
%A A118961 Lekraj Beedassy (blekraj(AT)yahoo.com), May 07 2006
%E A118961 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 11 2006

%I A114706
%S A114706 1,1,2,1,8,2,1,22,20,2,1,52,106,36,2,1,114,420,310,56,2,1,240,1410,1840,
%T A114706 706,80,2,1,494,4260,8714,5832,1382,108,2,1,1004,11978,35484,36898,
%U A114706 15100,2442,140,2,1,2026,31988,129758,194216,122674,34012,4006,176,2,1
%N A114706 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n and having k ascents (n>=1; 0<=k<=n-1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. An ascent in a Schroeder path is a maximal sequence of consecutive U steps.
%C A114706 Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..n-1)=2*A049608(n-1).
%F A114706 G.f.=G-1, where G=G(t, z) satisfies z(1+t-z+tz)G^2-(1+tz)G+1=0.
%e A114706 T(3,2)=2 because we have (U)H(U)DD and (UU)D(U)DD, where U=(1,1), D=(1,-1),
%e A114706 H=(2,0) (the ascents are shown between parentheses).
%e A114706 Triangle starts:
%e A114706 1;
%e A114706 1,2;
%e A114706 1,8,2;
%e A114706 1,22,20,2;
%p A114706 G:=(1+t*z-sqrt(1-2*t*z+t^2*z^2-4*z-4*z^2*t+4*z^2))/2/(z+t*z+z^2*t-z^2)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 11 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form
%Y A114706 Cf. A001003, A049608.
%Y A114706 Adjacent sequences: A114703 A114704 A114705 this_sequence A114707 A114708 A114709
%Y A114706 Sequence in context: A066532 A020778 A118961 this_sequence A046740 A130562 A011208
%K A114706 nonn,tabl
%O A114706 1,3
%A A114706 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2005

%I A046740
%S A046740 1,1,1,2,1,8,2,1,22,28,2,1,52,182,72,2,1,114,864,974,164,2,1,240,3474,
%T A046740 8444,4174,352,2,1,494,12660,57194,61464,15782,732,2,1,1004,43358,
%U A046740 332528,660842,373940,55286,1496,2,1,2026,142552,1747558,5814124
%N A046740 Triangle of number of permutations of [n] with 0 successions, by number of rises.
%C A046740 The recurrence given by Roselle is wrong.
%D A046740 D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.
%F A046740 a(n, 1) = 1; for r > 1, a(n, r)=r*a(n-1, r)+(n-r)*a(n-1, r-1)+(n-2)*a(n-2, r-1).
%F A046740 a(n, 2) = 2^n-2*n = 2*A000295 = A005803, n >= 3.
%e A046740 1; 1; 1 2; 1 8 2; 1 22 28 2; ...
%Y A046740 Cf. A046739, A000295. Row sums give A000255. Diagonals give A005803, A065340.
%Y A046740 Row sums give A000255.
%Y A046740 Adjacent sequences: A046737 A046738 A046739 this_sequence A046741 A046742 A046743
%Y A046740 Sequence in context: A020778 A118961 A114706 this_sequence A130562 A011208 A001281
%K A046740 nonn,easy,nice,tabf
%O A046740 1,4
%A A046740 njas
%E A046740 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 03 2003

%I A130562
%S A130562 1,2,1,8,2,2,16,8,4,6,128,16,16,4,24,256,128,32,16,48,120,1024,256,256,
%T A130562 32,192,240,720,2048,1024,512,256,384,64,96,5040,32768,2048,2048,512,
%U A130562 3072,384,384,10080,40320,65536,32768,4096,2048,6144,3072,2304,40320
%N A130562 Triangular table of denominators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).
%C A130562 The corresponding numerator table is given in A131440.
%H A130562 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 A130562 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. 775, 22.3.9.
%H A130562 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A130562.text">Rational coefficients and more. </a>
%F A130562 a(n,m)=denom(L(1/2,n,m)) with L(1/2,n,m)=((-1)^m)*binomial(n+1/2,n-m)/m!, n>=m>=0, else 0 (taken in lowest terms).
%e A130562 [1]; [2,1]; [8,2,2]; [16,8,4,6]; [128,16,16,4,24]; [256,128,32,16,48,120]; ...
%Y A130562 Cf. A021009 (Coefficient table of n!*L(n, 0, x).
%Y A130562 Adjacent sequences: A130559 A130560 A130561 this_sequence A130563 A130564 A130565
%Y A130562 Sequence in context: A118961 A114706 A046740 this_sequence A011208 A001281 A065826
%K A130562 nonn,tabl,easy
%O A130562 0,2
%A A130562 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jul 13 2007

%I A011208
%S A011208 1,1,2,1,8,2,8,3,9,6,2,5,4,0,0,2,2,7,0,2,9,3,8,7,3,9,0,2,7,3,5,1,5,
%T A011208 2,2,4,7,8,1,1,0,5,5,9,6,6,2,2,7,6,1,3,2,0,2,5,6,0,0,3,6,4,7,2,7,1,
%U A011208 0,7,0,4,1,5,9,3,5,3,3,8,4,9,3,5,0,1,6,9,5,6,4,0,6,5,5,4,2,6,7,4,4
%N A011208 Decimal expansion of 14th root of 5.
%Y A011208 Adjacent sequences: A011205 A011206 A011207 this_sequence A011209 A011210 A011211
%Y A011208 Sequence in context: A114706 A046740 A130562 this_sequence A001281 A065826 A123235
%K A011208 nonn,cons
%O A011208 1,3
%A A011208 njas

%I A001281
%S A001281 0,2,1,8,2,14,3,20,4,26,5,32,6,38,7,44,8,50,9,56,10,62,11,68,12,74,13,80,
%T A001281 14,86,15,92,16,98,17,104,18,110,19,116,20,122,21,128,22,134,23,140,24,
%U A001281 146,25,152,26,158,27,164,28,170,29,176,30,182,31,188,32,194,33
%N A001281 Image of n under the map n->n/2 if n even, n->3n-1 if n odd.
%C A001281 On the set of positive integers, the orbit of any number seems to end in the orbit of 1, of 5 or of 17. Writing n=1+q*2^p with q odd, it is easily seen that for p=0,1 and p>3, some iterations of the map lead to a strictly smaller number (for n>17). The cases p=2 and p=3 may give rise to bigger loops (depending on the form of q). See sequences A135727-A135729 for maxima of the orbits and corresponding record indices. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 29 2007
%D A001281 R. K. Guy, Unsolved Problems in Number Theory, E16.
%H A001281 J. C. Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html">The 3x+1 problem and its generalizations</a>, Amer. Math. Monthly, 92 (1985), 3-23.
%F A001281 f(n) = (7n-2-(5n-2)*cos(pi n))/4 [ Robert W. Craigen (craigen(AT)fresno.edu) ]
%p A001281 f := n-> if n mod 2 = 0 then n/2 else 3*n-1; fi;
%o A001281 (PARI) A001281(n)=if(n%2,3*n-1,n>>1) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 29 2007
%Y A001281 Cf. A037082.
%Y A001281 Cf. A037084, A039500-A039505, A135727-A135730. See also A006370, A006577 (Collatz 3x+1 problem).
%Y A001281 Adjacent sequences: A001278 A001279 A001280 this_sequence A001282 A001283 A001284
%Y A001281 Sequence in context: A046740 A130562 A011208 this_sequence A065826 A123235 A021462
%K A001281 nonn
%O A001281 0,2
%A A001281 njas

%I A065826
%S A065826 1,1,2,1,8,3,1,22,33,4,1,52,198,104,5,1,114,906,1208,285,6,1,240,3573,
%T A065826 9664,5955,720,7,1,494,12879,62476,78095,25758,1729,8,1,1004,43824,
%U A065826 352936,780950,529404,102256,4016,9,1,2026,143520,1820768,6551770
%N A065826 Triangle with T(n,k)=k*E(n,k) where E(n,k) are Eulerian numbers A008292.
%C A065826 Row sums are (n+1)!/2, i.e. A001710 offset, implying that if n balls are put at random into n boxes, the expected number of boxes with at least one ball is (n+1)/2, and the expected number of empty boxes is (n-1)/2.
%F A065826 T(n, k) = k*(a(n-1, k)+a(n-1, k-1)*(n-k+1)/(k-1)) [with T(n, 1) = 1] = Sum k*(-1)^j*(k-j)^n*C(n+1, j), j = 0..k.
%e A065826 Rows start: 1; 1,2; 1,8,3; 1,22,33,4; etc.
%Y A065826 Adjacent sequences: A065823 A065824 A065825 this_sequence A065827 A065828 A065829
%Y A065826 Sequence in context: A130562 A011208 A001281 this_sequence A123235 A021462 A082834
%K A065826 nonn,tabl
%O A065826 1,3
%A A065826 Henry Bottomley (se16(AT)btinternet.com), Dec 06 2001

%I A123235
%S A123235 1,1,1,2,1,8,3,1,48,16,7,384,120,59,1,3840,1152,606,17,46080,13440,7392,
%T A123235 263,1,645120,184320,104640,4288,31,10321920,2903040,1687680,76000,759,
%U A123235 1,185794560,51609600,30562560,1472640,17950,49
%V A123235 1,1,1,2,1,8,3,-1,48,16,-7,384,120,-59,1,3840,1152,-606,17,46080,13440,-7392,263,-1,
%W A123235 645120,184320,-104640,4288,-31,10321920,2903040,-1687680,76000,-759,1,185794560,
%X A123235 51609600,-30562560,1472640,-17950,49
%N A123235 Strange recursive polynomial triangular array related to the Bessel function has two levels for each power of the polynomial variable: p(k, x) = 2*(k - 1)*p(k - 1, x) -x*p(k - 2, x).
%C A123235 The Bessel recursive polynomial from Jahnke and Emde is: Z(p-1,x)+Z(p+1,x)=(2*p/x)*Z(p,x) Rearranging gives: x*Z(p+1,x)=2*p*Z(p,x)-x*Z(p-1,x) Replace p with k-1: x*Z(k,x)=2*(k-1)*Z(k-1,x)-x*Z(k-2,x) By doing it in two levels it appears that a Bessel polynomial results. Triangle row sums give A093856
%D A123235 Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York,1945, page 144
%F A123235 p(k, x) = 2*(k - 1)*p(k - 1, x) - x*p(k - 2, x)
%e A123235 {1},
%e A123235 {1, 1},
%e A123235 {2, 1},
%e A123235 {8, 3, -1},
%e A123235 {48, 16, -7},
%e A123235 {384, 120, -59, 1},
%e A123235 {3840, 1152, -606, 17},
%e A123235 {46080, 13440, -7392, 263, -1}
%t A123235 p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = 2*(k - 1)*p[k - 1, x] - x*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Floor[w]
%Y A123235 Cf. A093856.
%Y A123235 Adjacent sequences: A123232 A123233 A123234 this_sequence A123236 A123237 A123238
%Y A123235 Sequence in context: A011208 A001281 A065826 this_sequence A021462 A082834 A075647
%K A123235 uned,tabl,sign
%O A123235 1,4
%A A123235 Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 06 2006

%I A021462
%S A021462 0,0,2,1,8,3,4,0,6,1,1,3,5,3,7,1,1,7,9,0,3,9,3,0,1,3,1,0,0,4,3,6,6,
%T A021462 8,1,2,2,2,7,0,7,4,2,3,5,8,0,7,8,6,0,2,6,2,0,0,8,7,3,3,6,2,4,4,5,4,
%U A021462 1,4,8,4,7,1,6,1,5,7,2,0,5,2,4,0,1,7,4,6,7,2,4,8,9,0,8,2,9,6,9,4,3
%N A021462 Decimal expansion of 1/458.
%Y A021462 Adjacent sequences: A021459 A021460 A021461 this_sequence A021463 A021464 A021465
%Y A021462 Sequence in context: A001281 A065826 A123235 this_sequence A082834 A075647 A085470
%K A021462 nonn,cons
%O A021462 0,3
%A A021462 njas

%I A082834
%S A082834 2,1,8,3,4,6,0,0,8,1,2,2,9,6,9,1,8,1,6,3,4,0
%N A082834 Decimal expansion of the (finite) value of the sum_{ k >= 1, k has no digit equal to 5 in base 10 } 1/k.
%D A082834 Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
%D A082834 Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.
%e A082834 21.83460081229691816340...
%Y A082834 Cf. A002387, A024101, A082830, A082831, A082832, A082833, A082835, A082836, A082837, A082838, A082839.
%Y A082834 Adjacent sequences: A082831 A082832 A082833 this_sequence A082835 A082836 A082837
%Y A082834 Sequence in context: A065826 A123235 A021462 this_sequence A075647 A085470 A099379
%K A082834 nonn,cons,more,base
%O A082834 2,1
%A A082834 Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 14 2003

%I A075647
%S A075647 2,1,8,3,4,9,5,6,12,32,7,10,15,16,30,11,14,18,20,25,24,13,22,21,28,35,36,
%T A075647 77,17,26,27,40,45,42,49,96,19,34,33,44,50,48,56,64,72,23,38,39,52,55,
%U A075647 54,63,80,81,120,29,46,51,60,65,66,70,88,90,100,187,31,58,57,68,75,78
%N A075647 List of groups in A075643.
%e A075647 2; 1,8; 3,4,9; 5,6,12,32; 7,10,15,16,30; 11,14,18,20,25,24; ...
%o A075647 (PARI) used = vector(50000); A = vector(70); A[1] = 2; B = A; C = A; D = A; D[1] = 1; print(2); used[2] = 1; x = vector(70, i, i); for (n = 2, 70, s = 0; for (j = 1, n - 1, while (used[x[j]], x[j] += j); print1(x[j], " "); used[x[j]] = 1; s += x[j]; x[j] += j); while (used[x[n]], x[n] += n); i = x[n] + n*((s + x[n])%(n + 1)); while(used[i], i += n*(n + 1)); print(i); used[i] = 1; s += i; A[n] = x[1] - 1; B[n] = i; C[n] = s; D[n] = s/(n + 1)); (Wasserman)
%Y A075647 Cf. A075643, A075644, A075645, A075646.
%Y A075647 Adjacent sequences: A075644 A075645 A075646 this_sequence A075648 A075649 A075650
%Y A075647 Sequence in context: A123235 A021462 A082834 this_sequence A085470 A099379 A133214
%K A075647 nonn,tabl
%O A075647 1,1
%A A075647 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 30 2002
%E A075647 More terms from David Wasserman (dwasserm(AT)earthlink.net), Jan 22 2005
%E A075647 Leading term 2 was omitted; restored May 16 2007

%I A085470
%S A085470 2,1,8,3,5,41,31,11,33,286,344,250,63,279,2577,4418,4822,2423,489,2895,
%T A085470 28624,64891,93624,70501,28504,4785,35685,378317,1073889,1916161,
%U A085470 1925999,1169751,392971,56475,509985,5795682,19792118,41973586
%V A085470 2,-1,8,-3,-5,41,-31,11,-33,286,-344,250,-63,-279,2577,-4418,4822,-2423,489,-2895,
%W A085470 28624,-64891,93624,-70501,28504,-4785,-35685,378317,-1073889,1916161,-1925999,1169751,
%X A085470 -392971,56475,-509985,5795682,-19792118,41973586
%N A085470 Triangle of coefficients of powers of e^2 in numerators of sum_{k=1..infty} {1 / [1+k^2*pi^2]^n}.
%H A085470 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfiniteSeries.html">Infinite Series</a>
%e A085470 {2, -3 + 8*e^2 - e^4, 11 - 31*e^2 + 41*e^4 - 5*e^6}
%Y A085470 Adjacent sequences: A085467 A085468 A085469 this_sequence A085471 A085472 A085473
%Y A085470 Sequence in context: A021462 A082834 A075647 this_sequence A099379 A133214 A110107
%K A085470 sign,tabl
%O A085470 1,1
%A A085470 Eric Weisstein (eric(AT)weisstein.com), Jul 01, 2003

%I A099379
%S A099379 0,0,2,1,8,3,8,1,24,6,16,1,28,5,16,14,64,5,30,1,52,10,24,1,80,30,36,27,
%T A099379 60,7,58,1,160,14,44,26,96,7,40,28,144,9,62,1,92,57,48,1,208,14,110,32,
%U A099379 124,9,108,38,176,22,72,1,176,11,64,51,384,64,94,1,156,26,122,1,264,11
%N A099379 The real part of n', the arithmetic derivative for Gaussian integers.
%C A099379 Ufnarovski and Ahlander briefly mention this idea, but they do not pursue it because the derivative of Gaussian integers is not an extension of the arithmetic derivative of integers. Recall that every nonzero Gaussian integer has a unique factorization into the product of a unit (1, -1, i, -i) and powers of positive Gaussian primes (i.e. Gaussian primes a+bi with a>0 and b>=0). The derivative of all positive Gaussian primes is 1. The derivative of 0 or a unit is 0. The derivative of a product follows the Leibnitz rule (uv)' = uv' + vu'. Note that (-u)' = -(u') and (iu)' = i(u'). This definition of a derivative can be extended to fractions u/v, where u and v are Gaussian integers. Indeed, the Mathematica code shown here works with such fractions.
%H A099379 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b099379.txt">Table of n, a(n) for n=0..2048</a>
%H A099379 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%H A099379 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaussianInteger.html">Gaussian Integer</a>
%F A099379 If n = u Product p_i^e_i, where the p_i are positive Gaussian primes and u is a unit, then a(n) = n * Sum (e_i/p_i).
%e A099379 For n=5, the factorization into positive Gaussian integers is -i (1+2i) (2+i). Using the formula, the derivative is 5 (1/(1+2i) + 1/(2+i)) = 3-3i. Hence a(5) = 3.
%t A099379 di[0]=0; di[1]=0; di[ -1]=0; di[I]=0; di[ -I]=0; di[n_] := Module[{f, unt}, f=FactorInteger[n, GaussianIntegers->True]; unt=(Abs[f[[1, 1]]]==1); If[unt, f=Delete[f, 1]]; f=Transpose[f]; Plus@@(n*f[[2]]/f[[1]])]; Re[Table[di[n], {n, 0, 100}]]
%Y A099379 Cf. A003415 (arithmetic derivative of n), A099380 (imaginary part of the Gaussian-integer derivative of n).
%Y A099379 Adjacent sequences: A099376 A099377 A099378 this_sequence A099380 A099381 A099382
%Y A099379 Sequence in context: A082834 A075647 A085470 this_sequence A133214 A110107 A110446
%K A099379 nice,nonn
%O A099379 0,3
%A A099379 T. D. Noe (noe(AT)sspectra.com), Oct 14 2004

%I A133214
%S A133214 1,1,2,1,8,4,1,18,36,8,1,32,144,128,16,1,50,400,800,400,32,1,72,900,
%T A133214 3200,3600,1152,64,1,98,1764,9800,19600,14112,3136,128,1,128,3136,25088,
%U A133214 78400,100352,50176,8192,256,1,162,5184,56448,254016,508032,451584
%N A133214 Delannoy paths counted by number of weak peaks.
%C A133214 T(n,k) = number of Delannoy paths (A001850) of size n with k weak peaks. A (central) Delannoy path is a lattice path of upsteps U=(1,1), downsteps D=(1,-1) and horizontal steps H=(2,0) that starts at the origin and ends on the x-axis. Its size is #Us + #Hs. Thus a Delannoy path of size n ends at the point (2n,0). A weak peak is a UD or an H.
%H A133214 See Example 3 in Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/vol6.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
%F A133214 T(n,k) = 2^k binomial(n,k)^2. GF: Sum_{n>=k>=0}T(n,k) x^n y^k = 1/Sqrt[(1-x)^2 - 4*x*y*(1+x-x*y)]
%e A133214 Table begins
%e A133214 \ k.0...1....2....3....4....5
%e A133214 n\
%e A133214 0 |.1
%e A133214 1 |.1...2
%e A133214 2 |.1...8....4
%e A133214 3 |.1..18...36....8
%e A133214 4 |.1..32..144..128...16
%e A133214 5 |.1..50..400..800..400...32
%e A133214 T(2,1)=8 counts the paths UUDD, UDDU, UHD, DUUD, DUDU, DUH, DHU, HDU
%e A133214 because each contains a single UD or a single H but not both.
%Y A133214 Row sums are the central Delannoy numbers A001850.
%Y A133214 Adjacent sequences: A133211 A133212 A133213 this_sequence A133215 A133216 A133217
%Y A133214 Sequence in context: A075647 A085470 A099379 this_sequence A110107 A110446 A109979
%K A133214 nonn,tabl
%O A133214 0,3
%A A133214 David Callan (callan(AT)stat.wisc.edu), Dec 18 2007

%I A110107
%S A110107 1,1,2,1,8,4,1,26,28,8,1,88,136,80,16,1,330,600,512,208,32,1,1360,2636,
%T A110107 2768,1648,512,64,1,6002,11892,14024,10544,4832,1216,128,1,27760,55376,
%U A110107 69728,60768,35712,13312,2816,256,1,132690,265200,347072,332768,231232
%N A110107 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k return steps to the line y=x from the line y=x+1 or from the line y=x-1 (i.e. E steps from the line y=x+1 to the line y=x or N steps from the line y=x-1 to the line y=x; a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1)).
%C A110107 Row sums are the central Delannoy numbers (A001850). sum(k*T(n,k),k=0..n)=2*A110099(n).
%D A110107 R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.
%F A110107 G.f.=1/(1-z-2tzR), where R=1+zR+zR^2 is the g.f. of the large Schroeder numbers (A006318).
%e A110107 T(2,1)=8 because we have DN(E), DE(N), N(E)D, ND(E), NNE(E), E(N)D, ED(N) and EEN(N) (the return E or N steps are shown between parentheses).
%e A110107 Triangle begins:
%e A110107 1;
%e A110107 1,2;
%e A110107 1,8,4;
%e A110107 1,26,28,8;
%e A110107 1,88,136,80,16;
%p A110107 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1-z-2*t*z*R): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
%Y A110107 Cf. A001850, A110098, A110099.
%Y A110107 Adjacent sequences: A110104 A110105 A110106 this_sequence A110108 A110109 A110110
%Y A110107 Sequence in context: A085470 A099379 A133214 this_sequence A110446 A109979 A110171
%K A110107 nonn,tabl
%O A110107 0,3
%A A110107 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 11 2005

%I A110446
%S A110446 1,2,1,8,4,1,32,24,6,1,136,128,48,8,1,592,680,320,80,10,1,2624,3552,
%T A110446 2040,640,120,12,1,11776,18368,12432,4760,1120,168,14,1,53344,94208,
%U A110446 73472,33152,9520,1792,224,16,1,243392,480096,423936,220416,74592,17136
%N A110446 Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.
%C A110446 T(n,k) = number of Delannoy paths (A001850) of steps east(E), north(N) and diagonal (D) (i.e., northeast) from (0,0) to (n,n) containing k Ds not preceded by an E.
%F A110446 Gf. G(z, t)=Sum_{n>=k>=0}T(n, k)*z^n*t^k is given by G(z, t)= (1 - z(4 + 2*t) - z^2(4 - 4*t - t^2))^(-1/2)
%e A110446 Table begins
%e A110446 \ k...0....1....2....3....4....
%e A110446 n\
%e A110446 0 |...1
%e A110446 1 |...2....1
%e A110446 2 |...8....4....1
%e A110446 3 |..32...24....6....1
%e A110446 4 |.136..128...48....8....1
%e A110446 5 |.592..680..320...80...10....1
%e A110446 The paths ENDD, NDDE, DEND, DNDE, DDEN, DDNE each have 2 Ds not preceded by an E,
%e A110446 and so T(3,2)=6.
%Y A110446 Column k=0 is A006139.
%Y A110446 Adjacent sequences: A110443 A110444 A110445 this_sequence A110447 A110448 A110449
%Y A110446 Sequence in context: A099379 A133214 A110107 this_sequence A109979 A110171 A104988
%K A110446 nonn,tabl
%O A110446 0,2
%A A110446 David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

%I A109979
%S A109979 1,2,1,8,4,1,36,20,6,1,172,104,36,8,1,852,552,212,56,10,1,4324,2968,
%T A109979 1236,368,80,12,1,22332,16104,7164,2336,580,108,14,1,116876,87976,41372,
%U A109979 14512,3980,856,140,16,1,618084,483192,238356,88848,26372,6312,1204,176
%N A109979 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k (1,1)-steps on the line y=x (a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps (E=1,0), N=(0,1) and D(1,1)).
%C A109979 Row sums are the central Delannoy numbers (A001850). First column yields A109980 sum(k*T(n,k),k=0..n)=A001109(n).
%D A109979 R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.
%F A109979 G.f.=[tz-z+sqrt(1-6z+z^2)]/(1-6z+2tz^2-t^2*z^2).
%e A109979 T(2,1)=4 because we have DNE, DEN, NED, and END.
%e A109979 Triangle begins:
%e A109979 1;
%e A109979 2,1;
%e A109979 8,4,1;
%e A109979 36,20,6,1;
%p A109979 G:=(t*z-z+sqrt(1-6*z+z^2))/(1-6*z+2*t*z^2-t^2*z^2): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
%Y A109979 Cf. A001850, A109980, A001109.
%Y A109979 Adjacent sequences: A109976 A109977 A109978 this_sequence A109980 A109981 A109982
%Y A109979 Sequence in context: A133214 A110107 A110446 this_sequence A110171 A104988 A136225
%K A109979 nonn,tabl
%O A109979 0,2
%A A109979 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 06 2005

%I A110171
%S A110171 1,2,1,8,4,1,38,18,6,1,192,88,32,8,1,1002,450,170,50,10,1,5336,2364,912,
%T A110171 292,72,12,1,28814,12642,4942,1666,462,98,14,1,157184,68464,27008,9424,
%U A110171 2816,688,128,16,1,864146,374274,148626,53154,16722,4482,978,162,18,1
%N A110171 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps; a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1)).
%C A110171 Row sums are the central Delannoy numbers (A001850). T(n,0)=A002003(n) for n>=1. T(n,1)=A050146(n) for n>=1. Column k for k>=1 has g.f. z^k*R^(k-1)*g*(1+z*R), where R=1+zR+zR^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318) and g=1/sqrt(1-6z+z^2) is the g.f. of the central Delannoy numbers (A001850). sum(k*T(n,k),k=0..n)=A050151(n) (the partial sums of the central Delannoy numbers) = (1/2)*n*R(n), where R(n)=A006318(n) is the n-th large Schroeder number.
%D A110171 R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.
%F A110171 G.f.=(1+z+Q)/[Q(2-t+tz+tQ)], where Q=sqrt(1-6z+z^2).
%e A110171 T(2,1)=4 because we have NED, NENE, NEEN, and NDE.
%e A110171 Triangle starts:
%e A110171 1;
%e A110171 2,1;
%e A110171 8,4,1;
%e A110171 38,18,6,1;
%e A110171 192,88,32,8,1;
%p A110171 Q:=sqrt(1-6*z+z^2): G:=(1+z+Q)/Q/(2-t+t*z+t*Q): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
%Y A110171 Cf. A001850, A002003, A050146, A006318, A050151.
%Y A110171 Adjacent sequences: A110168 A110169 A110170 this_sequence A110172 A110173 A110174
%Y A110171 Sequence in context: A110107 A110446 A109979 this_sequence A104988 A136225 A089460
%K A110171 nonn,tabl
%O A110171 0,2
%A A110171 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 14 2005

%I A104988
%S A104988 1,2,1,8,4,1,42,20,6,1,266,120,38,8,1,1954,836,270,62,10,1,16270,6616,
%T A104988 2150,516,92,12,1,151218,58576,19030,4688,882,128,14,1,1551334,573672,
%U A104988 185674,46516,9050,1392,170,16,1,17414114,6159976,1982310,502324,99994
%N A104988 Matrix square of triangle A104980.
%C A104988 Triangular matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.
%F A104988 T(n+1, 0) = 2*A104980(n+3, 3) = 2*A104982(n) for n>=0.
%e A104988 Triangle begins:
%e A104988 1;
%e A104988 2,1;
%e A104988 8,4,1;
%e A104988 42,20,6,1;
%e A104988 266,120,38,8,1;
%e A104988 1954,836,270,62,10,1;
%e A104988 16270,6616,2150,516,92,12,1;
%e A104988 151218,58576,19030,4688,882,128,14,1;
%e A104988 1551334,573672,185674,46516,9050,1392,170,16,1;
%e A104988 17414114,6159976,1982310,502324,99994,15956,2070,218,18,1; ...
%o A104988 (PARI) {T(n,k)=if(n<k|k<0,0,(matrix(n+1,n+1,m,j,if(m==j,1,if(m==j+1,-m+1, -polcoeff((1-1/sum(i=0,m,i!*x^i))/x+O(x^m),m-j-1))))^-2)[n+1,k+1])}
%Y A104988 Cf. A104980, A104982 (column 0), A104989 (row sums).
%Y A104988 Adjacent sequences: A104985 A104986 A104987 this_sequence A104989 A104990 A104991
%Y A104988 Sequence in context: A110446 A109979 A110171 this_sequence A136225 A089460 A135520
%K A104988 nonn,tabl
%O A104988 0,2
%A A104988 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 10 2005

%I A136225
%S A136225 1,2,1,8,4,1,49,26,6,1,414,232,54,8,1,4529,2657,629,92,10,1,61369,37405,
%T A136225 9003,1320,140,12,1,996815,627435,153276,22606,2385,198,14,1,18931547,
%U A136225 12248365,3031553,450066,47500,3904,266,16,1,412345688,273211787
%N A136225 Matrix square of triangle A136220, read by rows.
%C A136225 Column 0 of this triangle = column 1 of square array A136217.
%F A136225 Let P=A136220, V=A136230, then column k of P^2 (this triangle) = column 0 of V^(k+1) while column j of V = column 0 of P^(3j+2).
%e A136225 Let P = A136220, then this triangle is P^2 and begins:
%e A136225 1;
%e A136225 2, 1;
%e A136225 8, 4, 1;
%e A136225 49, 26, 6, 1;
%e A136225 414, 232, 54, 8, 1;
%e A136225 4529, 2657, 629, 92, 10, 1;
%e A136225 61369, 37405, 9003, 1320, 140, 12, 1;
%e A136225 996815, 627435, 153276, 22606, 2385, 198, 14, 1;
%e A136225 18931547, 12248365, 3031553, 450066, 47500, 3904, 266, 16, 1; ...
%e A136225 where column k of P^2 = column 0 of V^(k+1) and
%e A136225 triangle V = A136230 begins:
%e A136225 1;
%e A136225 2, 1;
%e A136225 8, 5, 1;
%e A136225 49, 35, 8, 1;
%e A136225 414, 325, 80, 11, 1;
%e A136225 4529, 3820, 988, 143, 14, 1;
%e A136225 61369, 54800, 14696, 2200, 224, 17, 1; ...
%e A136225 where column k of V = column 0 of P^(3k+2).
%e A136225 Triangle P = A136220 begins:
%e A136225 1;
%e A136225 1, 1;
%e A136225 3, 2, 1;
%e A136225 15, 10, 3, 1;
%e A136225 108, 75, 21, 4, 1;
%e A136225 1036, 753, 208, 36, 5, 1;
%e A136225 12569, 9534, 2637, 442, 55, 6, 1;
%e A136225 185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
%e A136225 where column k of P = column 0 of U^(k+1) and U = A136228.
%o A136225 (PARI) {T(n,k)=local(P=Mat(1),U,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));U=P*PShR^2; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));(P^2)[n+1,k+1]}
%Y A136225 Cf. columns: A136226, A136227; related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.
%Y A136225 Adjacent sequences: A136222 A136223 A136224 this_sequence A136226 A136227 A136228
%Y A136225 Sequence in context: A109979 A110171 A104988 this_sequence A089460 A135520 A136230
%K A136225 nonn,tabl
%O A136225 0,2
%A A136225 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 28 2008

%I A089460
%S A089460 1,2,1,8,4,1,50,24,6,1,432,200,48,8,1,4802,2160,500,80,10,1,65536,28812,
%T A089460 6480,1000,120,12,1,1062882,458752,100842,15120,1750,168,14,1,20000000,
%U A089460 8503056,1835008,268912,30240,2800,224,16,1,428717762,180000000
%N A089460 Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.
%C A089460 Equals the matrix square of A088956 when treated as a lower triangular matrix. The 2nd hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = sum(k=0..n, T(n,k)*b(k)), where T(n,k) = 2*(n-k+2)^(n-k-1)*C(n,k). Given a table in which the n-th row is the n-th binomial transform of the first row, then the 2nd hyperbinomial transform of any diagonal results in the diagonal located 2 diagonals lower in the table.
%F A089460 T(n, k) = 2*(n-k+2)^(n-k-1)*C(n, k). E.g.f.: exp(x*y)*(-LambertW(-y)/y)^2. Note: (-LambertW(-y)/y)^2 = sum(n>=0, 2*(n+2)^(n-1)*y^n/n!).
%e A089460 Rows begin:
%e A089460 {1},
%e A089460 {2,1},
%e A089460 {8,4,1},
%e A089460 {50,24,6,1},
%e A089460 {432,200,48,8,1},
%e A089460 {4802,2160,500,80,10,1},
%e A089460 {65536,28812,6480,1000,120,12,1},
%e A089460 {1062882,458752,100842,15120,1750,168,14,1},..
%Y A089460 Cf. A089461(row sums), A089462(diagonal), A089463, A088956.
%Y A089460 Adjacent sequences: A089457 A089458 A089459 this_sequence A089461 A089462 A089463
%Y A089460 Sequence in context: A110171 A104988 A136225 this_sequence A135520 A136230 A004732
%K A089460 nonn,tabl
%O A089460 0,2
%A A089460 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 05 2003

%I A135520
%S A135520 2,1,8,4,32,16,128,64,512,256,2048,1024,8192,4096,32768,16384,131072,
%T A135520 65536,524288,262144,2097152,1048576,8388608,4194304,33554432,16777216,
%U A135520 134217728,67108864,536870912,268435456,2147483648,1073741824
%N A135520 a(n)=a(n-1)+4a(n-2)-4a(n-3).
%F A135520 O.g.f.: [ -5/(2x-1)+3/(2x+1)]/4. a(n) = [5*2^n+3*(-2)^n]/4. a(2n)=2*A000302(n). a(2n+1)=A000302(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), corrected Apr 14 2008
%Y A135520 Cf. A097163, A097164.
%Y A135520 Adjacent sequences: A135517 A135518 A135519 this_sequence A135521 A135522 A135523
%Y A135520 Sequence in context: A104988 A136225 A089460 this_sequence A136230 A004732 A011244
%K A135520 nonn
%O A135520 0,1
%A A135520 Paul Curtz (bpcrtz(AT)free.fr), Feb 19 2008
%E A135520 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 23 2008

%I A136230
%S A136230 1,2,1,8,5,1,49,35,8,1,414,325,80,11,1,4529,3820,988,143,14,1,61369,
%T A136230 54800,14696,2200,224,17,1,996815,932761,257264,39468,4123,323,20,1,
%U A136230 18931547,18426632,5198680,812801,86506,6919,440,23,1,412345688
%N A136230 Triangle V, read by rows, where column k of V^(j+1) = column j of P^(3k+2), for j>=0, k>=0, and where P=A136220.
%F A136230 Triangle W=P^3=A136231 transforms column k of V into column k+1 of V. This triangle equals the matrix products: V = P^2 * [P shift right one column] and V = U * [U shift down one row] (see examples).
%e A136230 This triangle V begins:
%e A136230 1;
%e A136230 2, 1;
%e A136230 8, 5, 1;
%e A136230 49, 35, 8, 1;
%e A136230 414, 325, 80, 11, 1;
%e A136230 4529, 3820, 988, 143, 14, 1;
%e A136230 61369, 54800, 14696, 2200, 224, 17, 1;
%e A136230 996815, 932761, 257264, 39468, 4123, 323, 20, 1;
%e A136230 18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1; ...
%e A136230 where column k of V = column 0 of P^(3k+2) and
%e A136230 triangle P = A136220 begins:
%e A136230 1;
%e A136230 1, 1;
%e A136230 3, 2, 1;
%e A136230 15, 10, 3, 1;
%e A136230 108, 75, 21, 4, 1;
%e A136230 1036, 753, 208, 36, 5, 1;
%e A136230 12569, 9534, 2637, 442, 55, 6, 1; ...
%e A136230 where column k of P^2 = column 0 of V^(k+1).
%e A136230 Also, this triangle V equals the matrix product:
%e A136230 V = P^2 * [P shift right one column]
%e A136230 where P^2 = A136225 begins:
%e A136230 1;
%e A136230 2, 1;
%e A136230 8, 4, 1;
%e A136230 49, 26, 6, 1;
%e A136230 414, 232, 54, 8, 1;
%e A136230 4529, 2657, 629, 92, 10, 1;
%e A136230 61369, 37405, 9003, 1320, 140, 12, 1; ...
%e A136230 and P shift right one column begins:
%e A136230 1;
%e A136230 0, 1;
%e A136230 0, 1, 1;
%e A136230 0, 3, 2, 1;
%e A136230 0, 15, 10, 3, 1;
%e A136230 0, 108, 75, 21, 4, 1;
%e A136230 0, 1036, 753, 208, 36, 5, 1; ...
%e A136230 Also, this triangle V equals the matrix product:
%e A136230 V = U * [U shift down one row]
%e A136230 where triangle U = A136228 begins:
%e A136230 1;
%e A136230 1, 1;
%e A136230 3, 4, 1;
%e A136230 15, 24, 7, 1;
%e A136230 108, 198, 63, 10, 1;
%e A136230 1036, 2116, 714, 120, 13, 1; ...
%e A136230 and U shift down one row begins:
%e A136230 1;
%e A136230 1, 1;
%e A136230 1, 1, 1;
%e A136230 3, 4, 1, 1;
%e A136230 15, 24, 7, 1, 1;
%e A136230 108, 198, 63, 10, 1, 1;
%e A136230 1036, 2116, 714, 120, 13, 1, 1; ...
%o A136230 (PARI) {T(n,k)=local(P=Mat(1),U=Mat(1),V=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1])))); U=P*PShR^2;V=P^2*PShR; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); V=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,V[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-2))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));V[n+1,k+1]}
%Y A136230 Cf. A136226 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136234 (V^2), A136237 (V^3); A136217, A136218.
%Y A136230 Adjacent sequences: A136227 A136228 A136229 this_sequence A136231 A136232 A136233
%Y A136230 Sequence in context: A136225 A089460 A135520 this_sequence A004732 A011244 A008517
%K A136230 nonn,tabl
%O A136230 0,2
%A A136230 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 28 2008

%I A004732
%S A004732 1,1,2,1,8,5,16,7,128,21,256,33,1024,429,2048,715,32768,
%T A004732 2431,65536,4199,262144,29393,524288,52003,4194304,185725,
%U A004732 8388608,334305,33554432,9694845,67108864,17678835,2147483648
%N A004732 Numerator of n!!/(n+3)!!.
%D A004732 S. Janson, On the traveling fly problem, Graph Theory Notes of New York, Vol. XXXI, 17, 1996.
%Y A004732 Cf. A004733.
%Y A004732 Adjacent sequences: A004729 A004730 A004731 this_sequence A004733 A004734 A004735
%Y A004732 Sequence in context: A089460 A135520 A136230 this_sequence A011244 A008517 A114193
%K A004732 nonn
%O A004732 0,3
%A A004732 njas

%I A011244
%S A011244 1,1,0,2,1,8,6,0,2,9,6,8,7,8,5,0,3,5,2,8,5,7,9,9,7,1,1,1,0,6,5,8,2,
%T A011244 1,7,9,1,6,2,1,9,8,9,3,2,3,6,3,4,3,6,7,8,0,3,2,7,8,7,9,3,5,8,3,4,3,
%U A011244 5,4,8,3,7,5,8,1,8,4,9,5,3,8,8,3,2,4,5,0,2,3,4,5,1,5,3,9,7,0,2,4,3
%N A011244 Decimal expansion of 20th root of 7.
%Y A011244 Adjacent sequences: A011241 A011242 A011243 this_sequence A011245 A011246 A011247
%Y A011244 Sequence in context: A135520 A136230 A004732 this_sequence A008517 A114193 A039683
%K A011244 nonn,cons
%O A011244 1,4
%A A011244 njas

%I A008517
%S A008517 1,1,2,1,8,6,1,22,58,24,1,52,328,444,120,1,114,1452,4400,3708,720,1,
%T A008517 240,5610,32120,58140,33984,5040,1,494,19950,195800,644020,785304,
%U A008517 341136,40320,1,1004,67260,1062500,5765500,12440064,11026296,3733920
%N A008517 Second-order Eulerian triangle T(n,k), 1<=k<=n.
%C A008517 When seen as coefficients of polynomials with descending exponents, evaluations are in A000311 (x=2) and A001662 (x=-1).
%C A008517 The row reversed triangle is A112007. There one can find comments on the o.g.f.s for the diagonals of the unsigned Stirling1 triangle |A008275|.
%C A008517 Stirling2(n,n-k) = sum(T(k,m+1)*binomial(n+k-1+m,2*k),m=0..k-1) k>=1. See the Graham et al. reference p. 257 eq. (6.43).
%C A008517 This triangle is the coefficient triangle of numerator polynomials appearing in the o.g.f. for the k-th diagonal of the Stirling2 triangle A048993.
%C A008517 The o.g.f. for column k satisfies the recurrence G(k,x)= x*(2*x*diff(G(k-1,x),x) + (2-k)*G(k-1,x))/(1-k*x),k>=2, with G(1,x)=1/(1-x). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005.
%C A008517 T(n,k) = 0 if n<k, T(1,1)=1, T(n,-1):=0, T(n,k)=k*T(n-1,k)+(2*n-k)*T(n-1,k-1).
%C A008517 This triangle is in some sense generated by the differential equation y' = 1-2/(1+x+y). (This is the differential equation satisfied by the function defined implicitly as x+y=exp(x-y). ) If we take y = a(0) + a(1)x + a(2)x^2 + a(3)x^3 +... and assume a(0)=c then all the a's may be calculated formaly in terms of c, and we have a(1)= (c-1)/(c+1) and for n>1 a(n) = 2^n/n! (1+c)^(1-2n)( T(n,1)c - T(n,2)c^2 + T(n,3)c^3...+ (-1)^(n-1) T(n,n)c^n ) - Moshe Newman (mshnoiman(AT)hotmail.com), Aug 08 2007
%D A008517 I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
%D A008517 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 256.
%D A008517 O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
%H A008517 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Second-OrderEulerianTriangle.html">Second-Order Eulerian Triangle</a>
%e A008517 1; 1,2; 1,8,6; 1,22,58,24; 1,52,328,444,120; ...
%o A008517 (PARI) T(n,k)=if(n<1,0,z=1+O(x); for(k=1,n,z=1+intformal(z^2*(z+y-1))); n!*polcoeff(polcoeff(z,n),k))
%Y A008517 Columns include A005803, A004301, A006260. Right-hand columns include A000142, A002538, A002539. Row sums are A001147.
%Y A008517 Adjacent sequences: A008514 A008515 A008516 this_sequence A008518 A008519 A008520
%Y A008517 Sequence in context: A136230 A004732 A011244 this_sequence A114193 A039683 A108084
%K A008517 nonn,tabl,nice,easy
%O A008517 1,3
%A A008517 njas
%E A008517 More terms from Michael Somos, Oct 13, 2002

%I A114193
%S A114193 1,2,1,8,6,1,40,36,10,1,224,224,80,14,1,1344,1440,600,140,18,1,8448,9504,
%T A114193 4400,1232,216,22,1,54912,64064,32032,10192,2184,308,26,1,366080,439296,232960,
%U A114193 81536,20160,3520,416,30,1,2489344,3055104,1697280,639744,176256,35904,5304,540
%V A114193 1,-2,1,8,-6,1,-40,36,-10,1,224,-224,80,-14,1,-1344,1440,-600,140,-18,1,8448,-9504,
%W A114193 4400,-1232,216,-22,1,-54912,64064,-32032,10192,-2184,308,-26,1,366080,-439296,232960,
%X A114193 -81536,20160,-3520,416,-30,1,-2489344,3055104,-1697280,639744,-176256,35904,-5304,540
%N A114193 Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x)), c(x) the g.f. of A000108.
%C A114193 Row sums are A114191. Diagonal sums are A114194. Inverse of A114192.
%F A114193 Riordan array ((sqrt(1+8x)-1)/(4x), (sqrt(1+8x)-1)^2/(16x)).
%F A114193 T(n, k) = (-2)^(n-k)*A039599(n, k) = (-2)^(n-k)*C(2*n, n-k)*(2*k+1)/(n+k+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2005
%e A114193 Triangle begins
%e A114193 1;
%e A114193 -2, 1;
%e A114193 8, -6, 1;
%e A114193 -40, 36, -10, 1;
%e A114193 224, -224, 80, -14, 1;
%e A114193 -1344, 1440, -600, 140, -18, 1;
%Y A114193 Adjacent sequences: A114190 A114191 A114192 this_sequence A114194 A114195 A114196
%Y A114193 Sequence in context: A004732 A011244 A008517 this_sequence A039683 A108084 A108085
%K A114193 easy,sign,tabl
%O A114193 0,2
%A A114193 Paul Barry (pbarry(AT)wit.ie), Nov 16 2005

%I A039683
%S A039683 1,2,1,8,6,1,48,44,12,1,384,400,140,20,1,3840,4384,1800,340,30,1,46080,
%T A039683 56448,25984,5880,700,42,1,645120,836352,420224,108304,15680,1288,56,1,
%U A039683 10321920,14026752,7559936,2153088,359184,36288,2184,72,1
%V A039683 1,-2,1,8,-6,1,-48,44,-12,1,384,-400,140,-20,1,-3840,4384,-1800,340,-30,1,
%W A039683 46080,-56448,25984,-5880,700,-42,1,-645120,836352,-420224,108304,-15680,
%X A039683 1288,-56,1,10321920,-14026752,7559936,-2153088,359184,-36288,2184,-72,1
%N A039683 Double Pochhammer triangle: expansion of x(x+2)(x+4)..(x+2n-2).
%C A039683 a(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
%D A039683 Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%H A039683 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A039683.text">First 9 rows and comment</a>.
%F A039683 a(n, m) = a(n-1, m-1) - 2*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((ln(1+2*x))/2)^m)/m!.
%F A039683 E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 11 2003
%F A039683 a(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
%e A039683 {1}, {2,1}, {8,6,1}, {48,44,12,1}, ...
%t A039683 Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]
%Y A039683 First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1). Cf. A051141, A051142.
%Y A039683 Adjacent sequences: A039680 A039681 A039682 this_sequence A039684 A039685 A039686
%Y A039683 Sequence in context: A011244 A008517 A114193 this_sequence A108084 A108085 A011135
%K A039683 sign,tabl
%O A039683 1,2
%A A039683 wouter.meeussen(AT)pandora.be
%E A039683 Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).

%I A108084
%S A108084 1,2,1,8,6,1,64,56,14,1,1024,960,280,30,1,32768,31744,9920,1240,62,1,
%T A108084 2097152,2064384,666624,89280,5208,126,1,268435456,266338304,87392256,
%U A108084 12094464,755904,21336,254,1,68719476736,68451041280,22638755840
%N A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).
%C A108084 For n>0, n-th row sum = Product(2^i+1,i=1..n) (i.e. A028362(n+1)).
%C A108084 Triangle T(n,k), 0<=k<=n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [[1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...]= A014236(first zero omitted)DELTA A077957 where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 23 2006
%F A108084 T(n,0)=2^(n*(n+1)/2)=A006125(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
%e A108084 Triangle begins:
%e A108084 1;
%e A108084 2, 1;
%e A108084 8, 6, 1;
%e A108084 64, 56, 14, 1;
%e A108084 1024, 960, 280, 30, 1;
%e A108084 32768, 31744, 9920, 1240, 62, 1;
%Y A108084 Cf. A028362.
%Y A108084 Adjacent sequences: A108081 A108082 A108083 this_sequence A108085 A108086 A108087
%Y A108084 Sequence in context: A008517 A114193 A039683 this_sequence A108085 A011135 A019816
%K A108084 nonn
%O A108084 0,2
%A A108084 Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jun 05 2005

%I A108085
%S A108085 1,2,1,8,6,1,64,56,14,1,1024,960,280,30,1,32768,31744,9920,1240,62,1,2097152,
%T A108085 2064384,666624,89280,5208,126,1,268435456,266338304,87392256,12094464,755904,
%U A108085 21336,254,1,68719476736,68451041280,22638755840,3183575040,205605888,6217920
%V A108085 1,2,-1,8,-6,1,64,-56,14,-1,1024,-960,280,-30,1,32768,-31744,9920,-1240,62,-1,2097152,
%W A108085 -2064384,666624,-89280,5208,-126,1,268435456,-266338304,87392256,-12094464,755904,
%X A108085 -21336,254,-1,68719476736,-68451041280,22638755840,-3183575040,205605888,-6217920
%N A108085 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) - T(n-1,k-1).
%C A108085 For n>0, n-th row sum = Product(2^i - 1), i=1..n (i.e. A005329(n)).
%C A108085 Triangle T(n,k), 0<=k<=n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [[ -1, 0, -2, 0, -4, 0, -8, 0, -16, 0, -32, 0, ...]= A014236(first zero omitted)DELTA -A077957 where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 23 2006
%Y A108085 Adjacent sequences: A108082 A108083 A108084 this_sequence A108086 A108087 A108088
%Y A108085 Sequence in context: A114193 A039683 A108084 this_sequence A011135 A019816 A082532
%K A108085 sign
%O A108085 0,2
%A A108085 Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jun 05 2005

%I A011135
%S A011135 2,1,8,6,7,2,4,1,4,7,8,8,6,5,5,6,1,1,2,7,3,7,5,2,8,8,9,6,0,7,1,2,8,
%T A011135 5,6,4,4,8,6,6,2,2,2,6,7,7,6,5,7,9,9,8,4,2,5,7,3,4,0,1,4,7,1,2,6,2,
%U A011135 4,0,8,2,7,4,3,9,6,5,5,6,0,8,8,3,6,3,7,7,3,0,9,8,6,1,3,6,2,0,6,1,0
%N A011135 Decimal expansion of 5th root of 50.
%Y A011135 Adjacent sequences: A011132 A011133 A011134 this_sequence A011136 A011137 A011138
%Y A011135 Sequence in context: A039683 A108084 A108085 this_sequence A019816 A082532 A049250
%K A011135 nonn,cons
%O A011135 1,1
%A A011135 njas

%I A019816
%S A019816 1,2,1,8,6,9,3,4,3,4,0,5,1,4,7,4,8,1,1,1,2,8,9,3,9,1,9,2,3,1,5,2,5,
%T A019816 1,7,6,0,1,3,2,3,5,6,4,6,4,7,1,4,6,8,7,2,0,9,2,7,0,4,8,8,7,3,9,7,7,
%U A019816 9,5,1,3,7,8,7,5,2,8,0,7,3,4,6,2,7,5,4,7,5,3,3,1,9,5,6,5,9,5,9,4,0
%N A019816 Decimal expansion of sine of 7 degrees.
%Y A019816 Adjacent sequences: A019813 A019814 A019815 this_sequence A019817 A019818 A019819
%Y A019816 Sequence in context: A108084 A108085 A011135 this_sequence A082532 A049250 A060587
%K A019816 nonn,cons
%O A019816 0,2
%A A019816 njas

%I A082532
%S A082532 1,2,1,8,7,4,17,14,9,2,23,16,7,34,25,14,1,36,23,8,49,34,17,64,47,28,7
%N A082532 a(n) = n^2-2*floor[n/sqrt(2)]^2.
%e A082532 a(3)=1 since 3^2-2*int[3/1.4142..]^2=9-2*2^2=9-8=1
%Y A082532 a(n)=1 for n in A001541.
%Y A082532 Adjacent sequences: A082529 A082530 A082531 this_sequence A082533 A082534 A082535
%Y A082532 Sequence in context: A108085 A011135 A019816 this_sequence A049250 A060587 A081800
%K A082532 easy,nonn
%O A082532 1,2
%A A082532 Carmine Suriano (surianonoi5(AT)libero.it), May 01 2003

%I A049250
%S A049250 2,1,8,7,6,3,1,3,2,1,0,11,11,2,11,11,3,9,1,11,11,11,7,3,11,2,8,4,3,5,
%T A049250 11,3,11,10,1,8,7,6,5,4,3,2,1,0,11,11,11,11,2,1,3,11,6,5,11,8,11,2,8,
%U A049250 11,11,3,9,11,11,11,7,3,11,11,11,11,11,1,11,4,11,2,1,11,2,11,1,11,11,6
%N A049250 Smallest nonnegative value taken on by 11x^2 - ny^2 for an infinite number of integer pairs (x, y).
%Y A049250 Adjacent sequences: A049247 A049248 A049249 this_sequence A049251 A049252 A049253
%Y A049250 Sequence in context: A011135 A019816 A082532 this_sequence A060587 A081800 A105672
%K A049250 nonn
%O A049250 1,1
%A A049250 David W. Wilson (davidwwilson(AT)comcast.net)

%I A060587
%S A060587 0,2,1,8,7,6,4,3,5,24,26,25,23,22,21,19,18,20,12,14,13,11,10,9,16,15,
%T A060587 17,72,74,73,80,79,78,76,75,77,69,71,70,68,67,66,64,63,65,57,59,58,56,
%U A060587 55,54,61,60,62,36,38,37,44,43,42,40,39,41,33,35,34,32,31,30,28,27,29
%N A060587 A ternary code: inverse of A060583.
%C A060587 Write n in base 3, then (working from left to right) if the k-th digit of n is equal to the digit to the left of it then this is the k-th digit of a(n), otherwise the k-th digit of a(n) is the element of {0,1,2} which has not just been compared, then read result as a base 3 number.
%H A060587 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A060587 a(n) =3a([n/3])+(-[n/3]-n mod 3) =3a([n/3])+A060588(n).
%e A060587 a(76) = 46 since 76 written in base 3 is 2211; this gives a first digit of 1( = 3-2-0) for a(n), a second digit of 2( = 2 = 2), a third digit of 0( = 3-1-2) and a fourth digit of 1( = 1 = 1); 1201 base 3 is 46.
%Y A060587 Adjacent sequences: A060584 A060585 A060586 this_sequence A060588 A060589 A060590
%Y A060587 Sequence in context: A019816 A082532 A049250 this_sequence A081800 A105672 A005489
%K A060587 base,nonn
%O A060587 0,2
%A A060587 Henry Bottomley (se16(AT)btinternet.com), Apr 04 2001

%I A081800
%S A081800 2,1,8,7,6,9,1,4,1,5,8,4,4,4,5,3
%N A081800 Decimal expansion of Bohr N1 velocity constant : V_n1=2.187691415844453.10^6 (m/sec).
%Y A081800 Adjacent sequences: A081797 A081798 A081799 this_sequence A081801 A081802 A081803
%Y A081800 Sequence in context: A082532 A049250 A060587 this_sequence A105672 A005489 A015152
%K A081800 cons,nonn
%O A081800 1,1
%A A081800 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 10 2003

%I A105672
%S A105672 1,2,1,8,7,8,1,2,1,26,25,26,19,20,19,26,25,26,1,2,1,8,7,8,1,2,1,80,79,
%T A105672 80,73,74,73,80,79,80,55,56,55,62,61,62,55,56,55,80,79,80,73,74,73,80,
%U A105672 79,80,1,2,1,8,7,8,1,2,1,26,25,26,19,20,19,26,25,26,1,2,1,8,7,8,1,2,1
%N A105672 a(1)=1 then letting f(n)=3^n for f(n)<k<=f(n+1) a(k)=f(n+1)-a(k-f(n)) (b(n,3) using pari code).
%F A105672 a(n+1)=1+sum(k=1, n, (-1)^k*(2-3*3^valuation(k, 3)))
%o A105672 (PARI) b(n,m)=if(n<2,1,m*m^floor(log(n-1)/log(m))-b(n-m^floor(log(n-1)/log(m)),m))
%Y A105672 Cf. A105669, A105670, A093347, A093348.
%Y A105672 Adjacent sequences: A105669 A105670 A105671 this_sequence A105673 A105674 A105675
%Y A105672 Sequence in context: A049250 A060587 A081800 this_sequence A005489 A015152 A021461
%K A105672 nonn
%O A105672 1,2
%A A105672 Benoit Cloitre (benoit7848c(AT)orange.fr), May 03 2005

%I A005489 M0181
%S A005489 1,1,2,1,8,7,32,31,96,97
%N A005489 Coefficients of order n in Baker-Campbell-Hausdorff expansion.
%D A005489 J. A. Oteo, The Baker-Campbell-Hausdorff formula and nested commutator relations, J. Math. Phys., 32 (1991), 419-421.
%Y A005489 Adjacent sequences: A005486 A005487 A005488 this_sequence A005490 A005491 A005492
%Y A005489 Sequence in context: A060587 A081800 A105672 this_sequence A015152 A021461 A127674
%K A005489 nonn,nice
%O A005489 1,3
%A A005489 David J. Thompson.

%I A015152
%S A015152 1,2,1,8,7,134,37,8516,12259,2196098,1896671,2242814912,8371908703,
%T A015152 9201070882046,50188906239587,150640766448080444,1945921801461187291,
%U A015152 9876134472777874272122,235305539337366453248999
%V A015152 1,2,1,8,7,134,37,8516,12259,2196098,-1896671,2242814912,8371908703,
%W A015152 9201070882046,-50188906239587,150640766448080444,1945921801461187291,
%X A015152 9876134472777874272122,-235305539337366453248999
%N A015152 Sum of (Gaussian) q-binomial coefficients for q=-2.
%Y A015152 Adjacent sequences: A015149 A015150 A015151 this_sequence A015153 A015154 A015155
%Y A015152 Sequence in context: A081800 A105672 A005489 this_sequence A021461 A127674 A075733
%K A015152 sign
%O A015152 0,2
%A A015152 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A021461
%S A021461 0,0,2,1,8,8,1,8,3,8,0,7,4,3,9,8,2,4,9,4,5,2,9,5,4,0,4,8,1,4,0,0,4,
%T A021461 3,7,6,3,6,7,6,1,4,8,7,9,6,4,9,8,9,0,5,9,0,8,0,9,6,2,8,0,0,8,7,5,2,
%U A021461 7,3,5,2,2,9,7,5,9,2,9,9,7,8,1,1,8,1,6,1,9,2,5,6,0,1,7,5,0,5,4,7,0
%N A021461 Decimal expansion of 1/457.
%Y A021461 Adjacent sequences: A021458 A021459 A021460 this_sequence A021462 A021463 A021464
%Y A021461 Sequence in context: A105672 A005489 A015152 this_sequence A127674 A075733 A123516
%K A021461 nonn,cons
%O A021461 0,3
%A A021461 njas

%I A127674
%S A127674 1,1,2,1,8,8,1,18,48,32,1,32,160,256,128,1,50,400,1120,1280,512,1,72,
%T A127674 840,3584,6912,6144,2048,1,98,1568,9408,26880,39424,28672,8192,1,128,
%U A127674 2688,21504,84480,180224,212992,131072,32768,1,162,4320,44352,228096
%V A127674 1,-1,2,1,-8,8,-1,18,-48,32,1,-32,160,-256,128,-1,50,-400,1120,-1280,512,1,-72,840,
%W A127674 -3584,6912,-6144,2048,-1,98,-1568,9408,-26880,39424,-28672,8192,1,-128,2688,-21504,
%X A127674 84480,-180224,212992,-131072,32768,-1,162,-4320,44352,-228096,658944,-1118208
%N A127674 Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros).
%D A127674 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).
%H A127674 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 A127674 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. 795.
%H A127674 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A127674.text">Row polynomials. </a>
%H A127674 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A127674 a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^(n-m))*(2^(2*m-1))*binomial(n+m,2*m)*(2*n)/(n+m).
%e A127674 [1];[ -1,2];[1,-8,8];[ -1,18,-48,32];[1,-32,160,-256,128];...
%e A127674 The T-polynomial for row n=3, [ -1,18,-48,32], is T(2*3,x)=-1*x + 18*x^2 -48*x^4 + 32*x^6.
%Y A127674 Cf. A075733 (different signs and offset). A084930 (coefficients of odd indexed T-polynomials).
%Y A127674 Cf. A053120 (coefficients of T-polynomials, with interspersed zeros).
%Y A127674 Adjacent sequences: A127671 A127672 A127673 this_sequence A127675 A127676 A127677
%Y A127674 Sequence in context: A005489 A015152 A021461 this_sequence A075733 A123516 A016446
%K A127674 sign,easy,tabl
%O A127674 0,3
%A A127674 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007

%I A075733
%S A075733 1,2,1,8,8,1,18,48,32,1,32,160,256,128,1,50,400,1120,1280,512,1,72,840,
%T A075733 3584,6912,6144,2048,1,98,1568,9408,26880,39424,28672,8192,1,128,2688,21504,
%U A075733 84480,180224,212992,131072,32768,1,162,4320,44352,228096,658944,1118208
%V A075733 1,-2,1,-8,8,1,-18,48,-32,1,-32,160,-256,128,1,-50,400,-1120,1280,-512,1,-72,840,
%W A075733 -3584,6912,-6144,2048,1,-98,1568,-9408,26880,-39424,28672,-8192,1,-128,2688,-21504,
%X A075733 84480,-180224,212992,-131072,32768,1,-162,4320,-44352,228096,-658944,1118208
%N A075733 Array of coefficients in Zagier's polynomials P_(n,0)(x).
%C A075733 Zagier's polynomial P_{n,m} (m<=n and m>=0) are used under pari-gp (for algorithmic acceleration of alternating or positive series)
%C A075733 Up to signs, coefficients of Chebyshev's T-polynomials for even index. See A127674.
%F A075733 P_(n, 0)(x) = (-1)^n*2^(2*n-1)*prod(i=1, n, x-cos(Pi*(2*i-1)/4/n)^2)
%Y A075733 Cf. A000332.
%Y A075733 Adjacent sequences: A075730 A075731 A075732 this_sequence A075734 A075735 A075736
%Y A075733 Sequence in context: A015152 A021461 A127674 this_sequence A123516 A016446 A086657
%K A075733 sign,tabl
%O A075733 1,2
%A A075733 Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 18 2002

%I A123516
%S A123516 1,2,1,8,8,3,48,72,54,15,384,768,864,480,105,3840,9600,14400,12000,5250,945,
%T A123516 46080,138240,259200,288000,189000,68040,10395,645120,2257920,5080320,7056000,
%U A123516 6174000,3333960,1018710,135135,10321920,41287680,108380160,180633600,197568000
%V A123516 1,2,-1,8,-8,3,48,-72,54,-15,384,-768,864,-480,105,3840,-9600,14400,-12000,5250,-945,
%W A123516 46080,-138240,259200,-288000,189000,-68040,10395,645120,-2257920,5080320,-7056000,
%X A123516 6174000,-3333960,1018710,-135135,10321920,-41287680,108380160,-180633600,197568000
%N A123516 Triangle read by rows: T(n,k)=(-1)^k*n!2^(n-2k)*binomial(n,k)binomial(2k,k) (0<=k<=n).
%C A123516 Row sums yield the double factorial numbers (A001147). T(n,0)=2^n*n!=A000165(n). T(n,n)=(-1)^n*A001147(n).
%D A123516 B. T. Gill, Math. Magazine, vol. 79, No. 4, 2006, p. 313, problem 1729.
%p A123516 T:=(n,k)->(-1)^k*n!*2^(n-2*k)*binomial(n,k)*binomial(2*k,k): for n from 0 to 8 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A123516 Cf. A001147, A000165.
%Y A123516 Adjacent sequences: A123513 A123514 A123515 this_sequence A123517 A123518 A123519
%Y A123516 Sequence in context: A021461 A127674 A075733 this_sequence A016446 A086657 A036296
%K A123516 sign,tabl
%O A123516 0,2
%A A123516 Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006

%I A016446
%S A016446 2,1,8,8,4,1,1,1,36,14,1,1,1,65,1,1,4,27,7,1,1,2,3,8,7,
%T A016446 4,1,7,1,4,13,1,3,1,2,3,6,1,4,2,2,1,1,2,8,2,6,2,1,5,3,1,
%U A016446 1,1,1,116,1,48,28,1,7,1,1,23,1,15,1,10,1,9,2,1,1,4,1,5
%N A016446 Continued fraction for ln(18).
%Y A016446 Adjacent sequences: A016443 A016444 A016445 this_sequence A016447 A016448 A016449
%Y A016446 Sequence in context: A127674 A075733 A123516 this_sequence A086657 A036296 A078105
%K A016446 nonn,cofr
%O A016446 1,1
%A