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%I #24 Sep 08 2022 08:45:49
%S 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,
%T 365,144,50,15,6,0,1,2867,1518,595,204,65,18,7,0,1,12240,6441,2511,
%U 858,270,81,21,8,0,1,53043,27774,10782,3672,1155,342,98,24,9,0,1
%N Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.
%H G. C. Greubel, <a href="/A171224/b171224.txt">Rows n = 0..100 of triangle, flattened</a>
%F 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.
%F 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
%F 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
%F 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
%e Triangle begins
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 3, 2, 0, 1;
%e 11, 6, 3, 0, 1;
%e 42, 23, 9, 4, 0, 1;
%e 167, 90, 36, 12, 5, 0, 1;
%e ...
%e Production array begins
%e 0, 1;
%e 1, 0, 1;
%e 3, 1, 0, 1;
%e 9, 3, 1, 0, 1;
%e 27, 9, 3, 1, 0, 1;
%e 81, 27, 9, 3, 1, 0, 1;
%e 243, 81, 27, 9, 3, 1, 0, 1;
%e ... - _Philippe Deléham_, Mar 04 2013
%t 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 *)
%o (Maxima)
%o 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 */
%o (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
%o (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
%o (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
%Y Cf. A053121, A097609, A065600.
%K nonn,tabl
%O 0,7
%A _Philippe Deléham_, Dec 05 2009
%E Terms a(55) onward added by _G. C. Greubel_, Apr 04 2019
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