A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.

1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 11, 6, 3, 0, 1, 42, 23, 9, 4, 0, 1, 167, 90, 36, 12, 5, 0, 1, 684, 365, 144, 50, 15, 6, 0, 1, 2867, 1518, 595, 204, 65, 18, 7, 0, 1, 12240, 6441, 2511, 858, 270, 81, 21, 8, 0, 1, 53043, 27774, 10782, 3672, 1155, 342, 98, 24, 9, 0, 1

Offset

0,7

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

Sum_{k=0..n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n+1), A002212(n+1), A026378(n+1) for x = 0, 1, 2, 3, 4 respectively.

Triangle equals B*A065600*B^(-1) = B^2*A097609*B^(-2) = B^3*A053121*B^(-3), product considered as infinite lower triangular arrays and B = A007318. - Philippe Deléham, Dec 08 2009

T(n,k) = T(n-1,k-1) + Sum_{i>=0} T(n-1,k+1+i)*3^i, T(0,0) = 1. - Philippe Deléham, Feb 23 2012

T(n,k) = ((k+1)/(n+1))*Sum_{j=0..floor((n-k)/2)} 3^(n-k-2*j)*C(n+1,j)*C(n-k-j-1,n-k-2*j)). - Vladimir Kruchinin, Apr 04 2019

Example

Triangle begins
    1;
    0,  1;
    1,  0,  1;
    3,  2,  0,  1;
   11,  6,  3,  0,  1;
   42, 23,  9,  4,  0,  1;
  167, 90, 36, 12,  5,  0,  1;
  ...
Production array begins
    0,  1;
    1,  0,  1;
    3,  1,  0,  1;
    9,  3,  1,  0,  1;
   27,  9,  3,  1,  0,  1;
   81, 27,  9,  3,  1,  0,  1;
  243, 81, 27,  9,  3,  1,  0,  1;
  ... - Philippe Deléham, Mar 04 2013

Mathematica

T[n_, k_]:= (k+1)/(n+1)*Sum[3^(n-k-2*j)*Binomial[n+1, j]*Binomial[n-k-j-1, n-k-2*j], {j, 0, Floor[(n-k)/2]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n} ]//Flatten (* G. C. Greubel, Apr 04 2019 *)

Prog

(Maxima)
T(n, k):=(k+1)/(n+1)*sum(3^(n-k-2*j)*binomial(n+1, j)*binomial(n-k-j-1, n-k-2*j), j, 0, floor((n-k)/2)); /* Vladimir Kruchinin, Apr 04 2019 */
(PARI) {T(n, k) = ((k+1)/(n+1))*sum(j=0, floor((n-k)/2), 3^(n-k-2*j) *binomial(n+1, j)*binomial(n-k-j-1, n-k-2*j))}; \\ G. C. Greubel, Apr 04 2019
(Magma) [[((k+1)/(n+1))*(&+[3^(n-k-2*j)*Binomial(n+1, j)*Binomial(n-k-j-1, n-k-2*j): j in [0..Floor((n-k)/2)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 04 2019
(Sage) [[((k+1)/(n+1))*sum(3^(n-k-2*j)*binomial(n+1, j)*binomial(n-k-j-1, n-k-2*j) for j in (0..floor((n-k)/2))) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 04 2019

Crossrefs

Cf. A053121, A097609, A065600.

Sequence in context: A193283 A193277 A118972 * A270741 A212220 A193233

Adjacent sequences: A171221 A171222 A171223 * A171225 A171226 A171227

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 05 2009

Extensions

Terms a(55) onward added by G. C. Greubel, Apr 04 2019

Status

approved