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A375780 a(n) = Sum_{k=0..n} binomial(n,k) * (k! * S(n,k))^2, where S(,) are Stirling numbers of second kind.

%I A375780 #16 Nov 07 2024 11:16:52
%S A375780 1,1,6,147,6940,536405,62352066,10136833063,2195583006072,
%T A375780 611230451090409,212649006828729790,90405046457569649531,
%U A375780 46115367523234055367828,27797472578675758999950013,19546873204979999617317371898,15858780455222184878234284613775,14703883436182303949571115531615216,15450188317599029331216704733732600017
%N A375780 a(n) = Sum_{k=0..n} binomial(n,k) * (k! * S(n,k))^2, where S(,) are Stirling numbers of second kind.
%H A375780 Paolo Xausa, <a href="/A375780/b375780.txt">Table of n, a(n) for n = 0..200</a>
%H A375780 MathsPower et al., <a href="https://mathoverflow.net/q/321324">Closed form for product of Stirling numbers of the second kind</a>, MathOverflow, 2019.
%F A375780 a(n) = n! * Sum_{k=0..n} k^n/k! * Sum_{m=0}^{n-k} (m+k)!/m!/(n-k-m)! * (-1)^m * S(n,m+k).
%F A375780 G.f.: the diagonal of 1 - t(x,y)*W'(t(x,y)), where t(x,y) := x*(1-e^y)*e^(x*(2-e^y)) and W() is Lambert W function.
%F A375780 a(n) ~ c * d^n * n^(2*n), where d = (2*r-1)^2*r/(exp(2)*(-1 + r + sqrt((1-r)*r))^2) = 0.522647981756854997298666108603651720918622906877425888529..., r = 0.665183670620587020892773716469052817866519211832581651... is the root of the equation (1-r)*(1 + r*LambertW(-1/(exp(1/r)*r)))^2 = r^3*LambertW(-1/(exp(1/r)*r))^2 and c = 1.38671243965876142096898080117513697606381035589463940412659515589... - _Vaclav Kotesovec_, Nov 07 2024
%t A375780 A375780[n_] := Sum[Binomial[n, k]*(k!*StirlingS2[n, k])^2, {k, 0, n}];
%t A375780 Array[A375780, 20, 0] (* _Paolo Xausa_, Nov 07 2024 *)
%o A375780 (PARI) { a375780(n) = sum(k=0,n, binomial(n,k) * (k!*stirling(n,k,2))^2); }
%Y A375780 Cf. A048144.
%K A375780 nonn
%O A375780 0,3
%A A375780 _Max Alekseyev_, Aug 27 2024