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 A384496 #9 May 31 2025 10:27:14
%S A384496 1,2,30,1044,68474,7180900,1050625720,196205015216,44361477901818,
%T A384496 11751610490415828,3567182462164189140,1220655384720089761080,
%U A384496 464932034143270233958352,195108754505934104188716064,89452431045403310104416682304,44489455448017524780072427344000
%N A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).
%C A384496 In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)) ~ 2^((m+2)*n + (m-3)/2) * n^(n - (m+1)/2) * w^(2*n) / (sqrt(m-1) * (w-1) * Pi^((m+1)/2) * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2).
%F A384496 a(n) ~ 2^(5*n - 1/2) * n^(n-2) * w^(2*n) / ((w-1) * Pi^2 * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166...
%t A384496 Table[Sum[Binomial[n,k]^3 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k,n-k]], {k, 0, n}], {n, 0, 20}]
%Y A384496 Cf. A187656 (m=0), A187658 (m=1), A384495 (m=2), A384472.
%K A384496 nonn
%O A384496 0,2
%A A384496 _Vaclav Kotesovec_, May 31 2025