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Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).
4

%I #17 Feb 05 2022 01:50:00

%S 1,1,5,41,468,6854,122582,2589978,63129392,1743732192,53827681152,

%T 1836453542472,68620052332752,2786929842106344,122241516227220504,

%U 5758920745460806824,290017142065771138560,15547326972257789803200,883974436758296523437760,53131928820278417749940544,3366145488853852112016117504

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

%H Seiichi Manyama, <a href="/A305981/b305981.txt">Table of n, a(n) for n = 0..376</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

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

%F a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n*(1 - exp(-1)) + 1/2)). - _Vaclav Kotesovec_, Aug 18 2018

%p 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

%t nmax = 20; CoefficientList[Series[1/(1 + LambertW[Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!

%t Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] k^k, {k, n}], {n, 20}]]

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k^k*stirling(n, k, 1)); \\ _Seiichi Manyama_, Feb 05 2022

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(log(1-x))))) \\ _Seiichi Manyama_, Feb 05 2022

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 18 2018