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 A382738 #19 Aug 30 2025 10:27:34
%S A382738 1,3,27,579,22779,1396803,121998267,14333812419,2175860165499,
%T A382738 414000255441603,96422983358827707,26970211126038920259,
%U A382738 8918364340126714711419,3440770498298077165166403,1531504734740033368269820347,778873986278207207346380124099
%N A382738 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n,k)^2.
%F A382738 a(n) == 0 (mod 3) for n > 0.
%F A382738 a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^3.
%F A382738 a(n) ~ sqrt(Pi) * n^(2*n + 5/2) / (16 * sqrt(1 - log(2)) * exp(2*n) * log(2)^(2*n+3)). - _Vaclav Kotesovec_, Aug 30 2025
%t A382738 Table[Sum[k! * (k+2)! * StirlingS2[n,k]^2/2, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 30 2025 *)
%o A382738 (PARI) a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 2)^2);
%Y A382738 Main diagonal of A382735.
%Y A382738 Cf. A048144, A382737, A382739.
%Y A382738 Cf. A382676.
%K A382738 nonn,changed
%O A382738 0,2
%A A382738 _Seiichi Manyama_, Apr 04 2025