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%I #7 Jun 29 2019 08:58:27
%S 1,1,6,57,736,11985,235296,5403937,142073856,4206560769,138483596800,
%T 5017244970441,198363105460224,8498001799768273,392127481640165376,
%U 19388814120804416625,1022681739669784231936,57317273018414456262273,3401527253966521309200384
%N Expansion of e.g.f. (1 + LambertW(-x))/(1 + 2*LambertW(-x)).
%F E.g.f.: 1 / (1 - Sum_{k>=1} k^k*x^k/k!).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^k * a(n-k).
%F a(n) ~ sqrt(Pi) * 2^(n - 3/2) * n^(n + 1/2) / exp(n/2). - _Vaclav Kotesovec_, Jun 29 2019
%t nmax = 18; CoefficientList[Series[(1 + LambertW[-x])/(1 + 2 LambertW[-x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%Y Cf. A000312, A002866, A086331, A218688, A277610, A308861, A308862.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 29 2019
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