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%I #20 Nov 14 2022 11:46:36
%S 1,1,6,51,614,9655,188209,4389532,119363488,3711190881,129932611723,
%T 5060364817200,217054300138136,10168837756846145,516709033266165479,
%U 28306732060349788908,1663231006737554997168,104344911495734048046929
%N E.g.f. satisfies A(x) = 1/(1 - x)^(exp(x) * A(x)).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.
%F E.g.f.: A(x) = exp( -LambertW(exp(x) * log(1-x)) ).
%F E.g.f.: A(x) = LambertW(exp(x) * log(1-x))/(exp(x) * log(1-x)).
%F a(n) ~ sqrt(1 + exp(1+r)/(1-r)) * n^(n-1) / (r^(n - 1/2) * exp(n-1)), where r = 0.249272970940807862774650581662931601739615720771408527... is the root of the equation exp(1+r) * log(1-r) = -1. - _Vaclav Kotesovec_, Nov 14 2022
%t nmax = 20; CoefficientList[Series[LambertW[E^x * Log[1-x]]/(E^x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 14 2022 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(exp(x)*log(1-x)))))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(lambertw(exp(x)*log(1-x))/(exp(x)*log(1-x))))
%Y Cf. A191365, A356926.
%Y Cf. A356752, A356753.
%Y Cf. A000272, A002104, A356927.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Sep 04 2022
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