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

1, 1, 6, 147, 6940, 536405, 62352066, 10136833063, 2195583006072, 611230451090409, 212649006828729790, 90405046457569649531, 46115367523234055367828, 27797472578675758999950013, 19546873204979999617317371898, 15858780455222184878234284613775, 14703883436182303949571115531615216, 15450188317599029331216704733732600017

Offset: 0

Author

Max Alekseyev, Aug 27 2024

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 0..200
• MathsPower et al., Closed form for product of Stirling numbers of the second kind, MathOverflow, 2019.

Crossrefs

Cf. A048144.

Programs

Mathematica

A375780[n_] := Sum[Binomial[n, k]*(k!*StirlingS2[n, k])^2, {k, 0, n}];
Array[A375780, 20, 0] (* Paolo Xausa, Nov 07 2024 *)

PARI

{ a375780(n) = sum(k=0,n, binomial(n,k) * (k!*stirling(n,k,2))^2); }

Formula

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).

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.

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