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 A362839 #17 May 05 2023 12:23:18
%S A362839 1,1,0,1,0,0,1,0,2,0,1,0,4,3,0,1,0,6,12,16,0,1,0,8,27,80,65,0,1,0,10,
%T A362839 48,216,560,336,0,1,0,12,75,448,2025,4512,1897,0,1,0,14,108,800,5120,
%U A362839 21708,40768,11824,0,1,0,16,147,1296,10625,67584,260253,407808,80145,0
%N A362839 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.
%F A362839 E.g.f. of column k: exp(x * (exp(k * x) - 1)).
%F A362839 G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).
%F A362839 T(n,k) = Sum_{j=0..n} (k*j-1)^(n-j) * binomial(n,j).
%e A362839 Square array begins:
%e A362839 1, 1, 1, 1, 1, 1, ...
%e A362839 0, 0, 0, 0, 0, 0, ...
%e A362839 0, 2, 4, 6, 8, 10, ...
%e A362839 0, 3, 12, 27, 48, 75, ...
%e A362839 0, 16, 80, 216, 448, 800, ...
%e A362839 0, 65, 560, 2025, 5120, 10625, ...
%o A362839 (PARI) T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);
%Y A362839 Columns k=0..3 give: A000007, A052506, A351736, A351737.
%Y A362839 Main diagonal gives A356806.
%Y A362839 Cf. A362652.
%K A362839 nonn,tabl
%O A362839 0,9
%A A362839 _Seiichi Manyama_, May 05 2023