This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A171224 #27 Oct 12 2024 01:41:22
%S A171224 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 A171224 365,144,50,15,6,0,1,2867,1518,595,204,65,18,7,0,1,12240,6441,2511,
%U A171224 858,270,81,21,8,0,1,53043,27774,10782,3672,1155,342,98,24,9,0,1
%N A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.
%H A171224 G. C. Greubel, <a href="/A171224/b171224.txt">Rows n = 0..100 of triangle, flattened</a>
%F A171224 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 A171224 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 A171224 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 A171224 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 A171224 Triangle begins
%e A171224 1;
%e A171224 0, 1;
%e A171224 1, 0, 1;
%e A171224 3, 2, 0, 1;
%e A171224 11, 6, 3, 0, 1;
%e A171224 42, 23, 9, 4, 0, 1;
%e A171224 167, 90, 36, 12, 5, 0, 1;
%e A171224 ...
%e A171224 Production array begins
%e A171224 0, 1;
%e A171224 1, 0, 1;
%e A171224 3, 1, 0, 1;
%e A171224 9, 3, 1, 0, 1;
%e A171224 27, 9, 3, 1, 0, 1;
%e A171224 81, 27, 9, 3, 1, 0, 1;
%e A171224 243, 81, 27, 9, 3, 1, 0, 1;
%e A171224 ... - _Philippe Deléham_, Mar 04 2013
%t A171224 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 A171224 (Maxima)
%o A171224 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 A171224 (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 A171224 (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 A171224 (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 A171224 Cf. A053121, A097609, A065600.
%K A171224 nonn,tabl
%O A171224 0,7
%A A171224 _Philippe Deléham_, Dec 05 2009
%E A171224 Terms a(55) onward added by _G. C. Greubel_, Apr 04 2019