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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..376
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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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
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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}]] -
PARI
a(n) = sum(k=0, n, (-1)^(n-k)*k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
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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)) + 1/2)). - Vaclav Kotesovec, Aug 18 2018