Search: id:A053121 Results 1-1 of 1 results found. %I A053121 %S A053121 1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,5,0,4,0,1,5,0,9,0,5,0,1,0,14,0,14,0,6, %T A053121 0,1,14,0,28,0,20,0,7,0,1,0,42,0,48,0,27,0,8,0,1,42,0,90,0,75,0,35,0,9, %U A053121 0,1,0,132,0,165,0,110,0,44,0,10,0,1,132,0,297,0,275,0,154,0,54,0,11,0 %N A053121 Catalan triangle (with 0's). Inverse lower triangular matrix of A049310(n, m) (coefficients of Chebyshev's S polynomials). %C A053121 "The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith] %C A053121 G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial). %C A053121 In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column of the inverse of a given Bell-matrix (here A049310) %C A053121 is obtained from its g.f. of the m=0 column (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference. %C A053121 Walks with a wall: triangle of number of n step walks from (0,0) to (n, m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant. %C A053121 Row sums of squares equals the Catalan sequence (A000108); for row 6: A000108(6) = 5^2 + 0^2 + 9^2 + 0^2 + 5^2 + 0^2 + 1^2 = 132. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 23 2005 %C A053121 Number of involutions of {1,2,...,n} that avoid the patterns 132 and have exactly k fixed points. Example: T(4,2)=3 because we have 2134, 4231 and 3214. Number of involutions of {1,2,...,n} that avoid the patterns 321 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1324 and 2134. Number of involutions of {1,2,...,n} that avoid the patterns 213 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1432 and 4231. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2006 %C A053121 Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007 %C A053121 This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1, k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0, 2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007 %C A053121 Riordan array (c(x^2),xc(x^2)), where c(x) is the g.f. of Catalan numbers A000108 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 25 2007 %C A053121 A053121^2 = triangle A145973. Convolved with A001405 = triangle A153585. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008] %C A053121 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 13 2009: (Start) %C A053121 By columns without the zeros, n-th row = A000108 convolved with itself %C A053121 n times; equivalent to A = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...), then %C A053121 n-th row = coefficients of A^(n+1). (End) %D A053121 I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785. %D A053121 J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002) %D A053121 E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 486 and 498). %D A053121 V. E. Hoggatt, Jr. and M. Bicknell, Catalan and Related Sequences Arising from Inverses of Pascal's Triangle Matrices, Fibonacci Quart. 14 (1976) 395-405. %D A053121 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fibonacci Quart. 38,5 (2000) 408-419; Note 4, pp. 414-415. %D A053121 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147. %D A053121 L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239. %D A053121 W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444. %H A053121 C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps %H A053121 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328. %H A053121 Index entries for sequences related to Chebyshev polynomials. %F A053121 a(n, m) := 0 if n0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m>0, a(n, m)=0 if m<0 - Henry Bottomley (se16(AT)btinternet.com), Jan 25 2001 %F A053121 Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even . - Philippe DELEHAM, May 26 2005 %F A053121 T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j, j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x); - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006 %F A053121 Sum_{k, 0<=k<=n}T(n,k)*(k+1)=2^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007 %F A053121 Sum_{j, j>=0}T(n,j)*binomial(j,k)= A054336(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007 %F A053121 T(2*n+1,2*k+1)=A039598(n,k), T(2*n,2*k)=A039599(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007 %F A053121 Sum_{k, 0<=k<=n} T(n,k)^x = A000027(n+1), A001405(n), A000108(n), A003161(n), A129123(n) for x = 0,1,2,3,4 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009] %e A053121 .......|...1 %e A053121 .......|.......1 %e A053121 .......|...1.......1 %e A053121 .......|.......2.......1 %e A053121 .......|...2.......3.......1 %e A053121 .......|.......5.......4.......1 %e A053121 .......|...5.......9.......5.......1 %e A053121 .......|......14......14.......6.......1 %e A053121 .......|..14......28......20.......7.......1 %e A053121 .......|......42......48......27.......8.......1 %e A053121 {1}; {0,1}; {1,0,1}; {0,2,0,1}; {2,0,3,0,1};... E.g. fourth row corresponds to polynomial p(3,x)= 2*x+x^3. %e A053121 Contribution from Paul Barry (pbarry(AT)wit.ie), May 29 2009: (Start) %e A053121 Production matrix is %e A053121 0, 1, %e A053121 1, 0, 1, %e A053121 0, 1, 0, 1, %e A053121 0, 0, 1, 0, 1, %e A053121 0, 0, 0, 1, 0, 1, %e A053121 0, 0, 0, 0, 1, 0, 1, %e A053121 0, 0, 0, 0, 0, 1, 0, 1, %e A053121 0, 0, 0, 0, 0, 0, 1, 0, 1, %e A053121 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End) %p A053121 T:=proc(n,k): if n+k mod 2 = 0 then (k+1)*binomial(n+1,(n-k)/2)/(n+1) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2006 %Y A053121 Cf. A008315, A049310, A033184, A000108, A001405. Another version: A008313. %Y A053121 Variant without zero-diagonals: A033184 and with rows reversed: A009766. %Y A053121 A145973, A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008] %Y A053121 Sequence in context: A125921 A029299 A049803 this_sequence A113408 A022337 A025687 %Y A053121 Cf. A108786 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009] %Y A053121 Adjacent sequences: A053118 A053119 A053120 this_sequence A053122 A053123 A053124 %K A053121 easy,nice,tabl,nonn,new %O A053121 0,8 %A A053121 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Search completed in 0.002 seconds