A355993 Expansion of e.g.f. -LambertW(x^3 * log(1-x))/6.

0, 0, 0, 0, 4, 10, 40, 210, 8064, 70560, 640800, 6375600, 189383040, 3165402240, 48879754560, 762766804800, 21652937349120, 525738717504000, 11796584629939200, 259139188966694400, 7842638783736115200, 240231375437935795200, 7066934411387842252800

Offset: 0

Author

Seiichi Manyama, Sep 24 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A052807, A355874, A357265.

Programs

Mathematica

With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3 * Log[1 - x]]/6, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3*log(1-x)))/6))

PARI

a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!)/6;

Formula

a(n) = (n!/6) * Sum_{k=1..floor(n/4)} k^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.