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

0,3

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.

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

Mathematica

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

Prog

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

Crossrefs

Cf. A048144.

Sequence in context: A358800 A063419 A065986 * A199226 A012791 A347842

Adjacent sequences: A375777 A375778 A375779 * A375781 A375782 A375783

Keyword

nonn

Author

Max Alekseyev, Aug 27 2024

Status

approved