A305981 Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).

1, 1, 5, 41, 468, 6854, 122582, 2589978, 63129392, 1743732192, 53827681152, 1836453542472, 68620052332752, 2786929842106344, 122241516227220504, 5758920745460806824, 290017142065771138560, 15547326972257789803200, 883974436758296523437760, 53131928820278417749940544, 3366145488853852112016117504

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..376

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

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)) + 1/2)). - Vaclav Kotesovec, Aug 18 2018

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[Abs[StirlingS1[n, k]] k^k, {k, n}], {n, 20}]]

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*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

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305819.

Sequence in context: A049119 A367423 A332236 * A032188 A240996 A346982

Adjacent sequences: A305978 A305979 A305980 * A305982 A305983 A305984

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 18 2018

Status

approved