A355874 Expansion of e.g.f. -LambertW(x^2 * log(1-x))/2.

0, 0, 0, 3, 6, 20, 450, 3024, 21840, 449280, 5690160, 68579280, 1491462720, 27798076800, 485405784864, 11821894207200, 285057334598400, 6578025489584640, 183420564173141760, 5342163886869062400, 152988752430721267200, 4897735504358795965440

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..415

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

Formula

a(n) = (n!/2) * Sum_{k=1..floor(n/3)} k^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

a(n) ~ c * n^(n-1) / (2 * exp(n) * r^n), where r = 0.6181555791782259971637080007872096609874188426179... is the root of the equation r^2 * log(1-r) = -exp(-1) and c = (sqrt(2 + exp(1)*r^3/(1-r))) = 1.91872811548413716811307748684660958454...

Mathematica

With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2 * Log[1 - x]]/2, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2*log(1-x)))/2))
(PARI) a(n) = n!*sum(k=1, n\3, k^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/2;

Crossrefs

Cf. A052807, A355993, A357265.

Cf. A355179.

Sequence in context: A355994 A375698 A356752 * A356967 A370996 A328567

Adjacent sequences: A355871 A355872 A355873 * A355875 A355876 A355877

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2022

Status

approved