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 A384472 #9 May 30 2025 10:53:41
%S A384472 1,2,22,558,25506,1770300,166190354,19647687682,2798281247682,
%T A384472 466166725448544,88942246964278060,19127775950813311232,
%U A384472 4578817457796314714502,1207681779462031251096888,348018457509475159702959174,108798555057988053563408904750,36676526343321856806298038370210
%N A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).
%C A384472 In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) ~ 2^((m+1)*n + (m-1)/2) * n^(n-(m+1)/2) / (sqrt(m-1) * Pi^((m+1)/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775.
%F A384472 a(n) ~ 2^(4*n + 1/2) * n^(n-2) / (Pi^2 * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...
%t A384472 Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}]
%Y A384472 Cf. A187655 (m=0), A187657 (m=1), A384471 (m=2), A384470.
%Y A384472 Cf. A226775.
%K A384472 nonn
%O A384472 0,2
%A A384472 _Vaclav Kotesovec_, May 30 2025