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%I #12 Sep 01 2020 05:28:59
%S 1,1,2,9,70,760,10451,173159,3350796,74100408,1842574557,50873252287,
%T 1543925883754,51078021158476,1829361243438535,70512079627989757,
%U 2910210706666701048,128046344157824920272,5982882723357716484777,295846337542679153759691,15435034787110130135765446,847314120718756196303796884,48820553176413784195229252939,2945834523040528378700506066289
%N E.g.f.: (1 + LambertW(-x))^(x/LambertW(-x)).
%C Logarithm of e.g.f. yields the e.g.f. of A304866.
%H Vaclav Kotesovec, <a href="/A300014/b300014.txt">Table of n, a(n) for n = 0..385</a>
%F E.g.f. A(x) = Sum_{n>=0} a(n) * x^n / n! satisfies:
%F (1) Sum_{n>=0} (n*x - log(A(x)))^n / n! = 1.
%F (2) Sum_{n>=0} (n*x - p*log(A(x)))^n / n! = (1 + LambertW(-x))^(p-1).
%F (3) Sum_{n>=0} ((n + m)*x - log(A(x)))^n / n! = ( LambertW(-x)/(-x) )^m.
%F (4) Sum_{n>=0} ((n + m)*x - p*log(A(x)))^n / n! = ( LambertW(-x)/(-x) )^m * (1 + LambertW(-x))^(p-1).
%F a(n) ~ 2^(1/2 - exp(-1)/2) * sqrt(Pi) * n^(n + exp(-1)/2 - 1/2) / Gamma(exp(-1)/2) * (1 + sqrt(2)*log(n)*Gamma(exp(-1)/2 + 1) / (sqrt(n)*Gamma(exp(-1)/2 - 1/2))). - _Vaclav Kotesovec_, Sep 01 2020
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 70*x^4/4! + 760*x^5/5! + 10451*x^6/6! + 173159*x^7/7! + 3350796*x^8/8! + 74100408*x^9/9! + 1842574557*x^10/10! + ...
%e RELATED SERIES.
%e log(A(x)) = x + x^2/2! + 5*x^3/3! + 40*x^4/4! + 434*x^5/5! + 5921*x^6/6! + 97152*x^7/7! + 1861224*x^8/8! + 40757712*x^9/9! + ... + A304866(n)*x^n/n! + ...
%e such that Sum_{n>=0} (n*x - log(A(x)))^n / n! = 1.
%t nmax = 20; CoefficientList[Series[(1 + LambertW[-x])^(x/LambertW[-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 01 2020 *)
%o (PARI) {a(n) = my(W = serreverse(-x*exp(x +x*O(x^n)))); n!*polcoeff( (1 + W)^(x/W), n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A304866.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 18 2018
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