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A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified September 30 11:00 EDT 2022. Contains 357105 sequences. (Running on oeis4.)