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 A382853 #15 Apr 07 2025 09:26:19
%S A382853 1,1,14,588,51064,7542780,1688795184,532244030976,224335607135616,
%T A382853 121793234373123840,82750681453274478720,68773648886955417943296,
%U A382853 68628724852793337500166144,80970628401965472953705395200,111490683570184861858636405923840,177177650274516448010905794637332480
%N A382853 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (k! * Stirling1(n,k))^2.
%F A382853 a(n) == 0 (mod n) for n > 0.
%F A382853 a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^n.
%F A382853 a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^n.
%F A382853 a(n) ~ c * (r*(1+r) + sqrt(r*(1+r)))^(2*n) * n^(2*n) / (exp(2*n) * r^n), where r = 0.71197519729041875298209529969157574831688314013967... is the root of the equation (1+r)*(r + LambertW(-1, -r*exp(-r)))^2 = r and c = 0.61294561390083215776201123658816241786650851195222... - _Vaclav Kotesovec_, Apr 07 2025
%t A382853 Table[Sum[Binomial[n+k-1,k] * k!^2 * StirlingS1[n,k]^2, {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Apr 07 2025 *)
%o A382853 (PARI) a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 1))^2);
%Y A382853 Cf. A382792, A382804, A382806.
%K A382853 nonn
%O A382853 0,3
%A A382853 _Seiichi Manyama_, Apr 06 2025