A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

1, 1, 3, 17, 132, 1334, 16442, 239994, 4041776, 77183328, 1647541632, 38877352392, 1004869488048, 28234217634024, 856830099396840, 27930093941047464, 973269467390922240, 36104568839480990400, 1420556927968241858880, 59088101641333114906944, 2590680379402887359111424

Offset: 0

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

• nonn

Comments

Inverse Stirling transform of A000312.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..397
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305981.

Programs

Maple

a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]

PARI

a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ Seiichi Manyama, Feb 05 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k^k.

a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - Vaclav Kotesovec, Aug 18 2018