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 A378320 #12 Nov 23 2024 09:27:35
%S A378320 1,1,0,1,1,0,1,2,1,0,1,3,3,2,0,1,4,6,6,3,0,1,5,10,13,11,6,0,1,6,15,24,
%T A378320 27,22,11,0,1,7,21,40,55,57,44,22,0,1,8,28,62,100,124,121,90,44,0,1,9,
%U A378320 36,91,168,241,278,258,187,90,0,1,10,45,128,266,432,570,620,555,392,187,0
%N A378320 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(2*n-2*r+k,r) * binomial(r,n-r)/(2*n-2*r+k) for k > 0.
%F A378320 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x^2 * A_k(x)^(2/k) )^k for k > 0.
%F A378320 G.f. of column k: B(x)^k where B(x) is the g.f. of A007477.
%F A378320 B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x^2 * B(x)^(k+1). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-2,k+1) for n > 1.
%e A378320 Square array begins:
%e A378320 1, 1, 1, 1, 1, 1, 1, ...
%e A378320 0, 1, 2, 3, 4, 5, 6, ...
%e A378320 0, 1, 3, 6, 10, 15, 21, ...
%e A378320 0, 2, 6, 13, 24, 40, 62, ...
%e A378320 0, 3, 11, 27, 55, 100, 168, ...
%e A378320 0, 6, 22, 57, 124, 241, 432, ...
%e A378320 0, 11, 44, 121, 278, 570, 1077, ...
%o A378320 (PARI) T(n, k, t=0, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
%o A378320 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378320 Columns k=0..1 give A000007, A007477.
%Y A378320 Cf. A071919, A378321.
%K A378320 nonn,tabl
%O A378320 0,8
%A A378320 _Seiichi Manyama_, Nov 23 2024