A355994 Expansion of e.g.f. -LambertW(x^2/2 * log(1-x)).

0, 0, 0, 3, 6, 20, 270, 1764, 12600, 169560, 1937880, 22300740, 349806600, 5556245760, 89073856872, 1678920566400, 33550354656000, 687175528253760, 15462823882213440, 370285712520237360, 9180722384533375200, 242398467521271149760

Offset

0,4

Links

Table of n, a(n) for n=0..21.

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

Formula

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

a(n) ~ sqrt(exp(r)/2 + r - 1/2) * n^(n-1) / (exp(n - 1/2) * (1 - exp(-r))^(n-1)), where r = 1.345291829921667620776408101642407390842578457922... is the root of the equation exp(1-2*r) - 2*exp(1-r) + exp(1) = 2/r. - Vaclav Kotesovec, Jan 31 2026

Mathematica

With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2/2 * Log[1 - x]], {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^2/2*log(1-x)))))
(PARI) a(n) = n!*sum(k=1, n\3, k^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

Crossrefs

Cf. A052807, A355995, A357265.

Sequence in context: A351492 A368165 A351505 * A375698 A356752 A355874

Adjacent sequences: A355991 A355992 A355993 * A355995 A355996 A355997

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2022

Status

approved