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 A382737 #23 May 05 2025 06:37:05
%S A382737 1,2,14,254,8654,467102,36414734,3862847774,534433092494,
%T A382737 93409669590302,20117959360842254,5233190283794276894,
%U A382737 1617259866279958581134,585633786711715561283102,245587300036701328750786574,118067003149791582488105955614,64502003996859329263691323378574
%N A382737 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling2(n,k)^2.
%H A382737 Robert Israel, <a href="/A382737/b382737.txt">Table of n, a(n) for n = 0..236</a>
%F A382737 a(n) == 0 (mod 2) for n > 0.
%F A382737 a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^2.
%F A382737 a(n) ~ sqrt(Pi) * n^(2*n + 3/2) / (4 * sqrt(1 - log(2)) * exp(2*n) * log(2)^(2*n+2)). - _Vaclav Kotesovec_, Apr 13 2025
%p A382737 f:= proc(n) local k; add(k!*(k+1)!*Stirling2(n,k)^2, k=0..n) end proc:
%p A382737 map(f, [$0..40]);
%t A382737 Table[Sum[k! * (k+1)! * StirlingS2[n,k]^2, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Apr 13 2025 *)
%o A382737 (PARI) a(n) = sum(k=0, n, k!*(k+1)!*stirling(n, k, 2)^2);
%Y A382737 Main diagonal of A382734.
%Y A382737 Cf. A048144, A382738, A382739.
%K A382737 nonn
%O A382737 0,2
%A A382737 _Seiichi Manyama_, Apr 04 2025