A355179 Expansion of e.g.f. -LambertW(x^2 * (1 - exp(x)))/2.

0, 0, 0, 3, 6, 10, 375, 2541, 11788, 317556, 4238685, 37921015, 909616026, 18283276518, 261259582675, 6360432558585, 164704011195480, 3332419310132776, 88606184592031353, 2713050497589230763, 71412977041725823750, 2144089948615678382970

Offset: 0

Author

Seiichi Manyama, Sep 24 2022

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A048802, A355180, A357267.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[(-LambertW[x^2 (1-Exp[x])])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 07 2025 *)

PARI

my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2*(1-exp(x)))/2)))

PARI

a(n) = n!*sum(k=1, n\3, k^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!)/2;

Formula

a(n) = (n!/2) * Sum_{k=1..floor(n/3)} k^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.