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 A378239 #34 Jul 09 2025 05:07:21
%S A378239 1,1,0,1,2,0,1,4,12,0,1,6,28,100,0,1,8,48,248,968,0,1,10,72,452,2480,
%T A378239 10208,0,1,12,100,720,4680,26688,113792,0,1,14,132,1060,7728,51504,
%U A378239 301648,1318832,0,1,16,168,1480,11800,87104,591312,3531424,15732064,0
%N A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2*n+2*r+k,n)/(2*n+2*r+k) for k > 0.
%F A378239 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
%F A378239 G.f. of column k: B(x)^k where B(x) is the g.f. of A219534.
%F A378239 B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-1,k+3) for n > 0.
%e A378239 Square array begins:
%e A378239 1, 1, 1, 1, 1, 1, 1, ...
%e A378239 0, 2, 4, 6, 8, 10, 12, ...
%e A378239 0, 12, 28, 48, 72, 100, 132, ...
%e A378239 0, 100, 248, 452, 720, 1060, 1480, ...
%e A378239 0, 968, 2480, 4680, 7728, 11800, 17088, ...
%e A378239 0, 10208, 26688, 51504, 87104, 136352, 202560, ...
%e A378239 0, 113792, 301648, 591312, 1017184, 1621280, 2454256, ...
%o A378239 (PARI) T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
%o A378239 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378239 Columns k=0..4 give A000007, A219534, A371693, A378155, A378156.
%Y A378239 Cf. A033877, A071949, A378236, A378237, A378238, A378240.
%K A378239 nonn,tabl
%O A378239 0,5
%A A378239 _Seiichi Manyama_, Nov 20 2024 based on suggestions from _Mikhail Kurkov_