A357265 Expansion of e.g.f. -LambertW(x * log(1-x)).

0, 0, 2, 3, 32, 150, 1884, 16380, 249808, 3255336, 59596560, 1037413080, 22432698144, 486784686960, 12233449250736, 316660035739320, 9111729094222080, 273147758526888000, 8880267446524694016, 301952732236006556160, 10963551960785051470080

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

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

Formula

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

a(n) ~ c * n^(n-1) / (exp(n) * r^n), where r = 0.51231128552386437348670059148148... is the root of the equation r*log(1-r) = -exp(-1) and c = sqrt(1 + exp(1)*r^2/(1-r)) = 1.56936827475369243605279148406... - Vaclav Kotesovec, Jan 25 2026

Mathematica

Table[n! * Sum[k^(k-1) * Abs[StirlingS1[n-k, k]]/(n-k)!, {k, 1, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 25 2026 *)

Prog

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

Crossrefs

Cf. A052807, A355842, A357267.

Sequence in context: A141049 A354612 A052830 * A356904 A041895 A371117

Adjacent sequences: A357262 A357263 A357264 * A357266 A357267 A357268

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2022

Status

approved