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E.g.f.: -log(1 + LambertW(-x) * (2 + LambertW(-x)) / 2).
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%I #8 Feb 16 2020 05:31:55

%S 1,2,8,49,409,4356,56734,877094,15742521,322454800,7434673036,

%T 190792267128,5398552673617,167087263076384,5617979017621650,

%U 203987454978218416,7957053981454827601,331920300203780633856,14746208516909980554736,695208730205550274544000

%N E.g.f.: -log(1 + LambertW(-x) * (2 + LambertW(-x)) / 2).

%F E.g.f.: -log(1 - Sum_{k>=1} k^(k-2) * x^k / k!).

%F a(n) = n^(n-2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * (n-k)^(n-k-2) * k * a(k).

%F a(n) ~ 2 * n^(n-2). - _Vaclav Kotesovec_, Feb 16 2020

%t nmax = 20; CoefficientList[Series[-Log[1 + LambertW[-x] (2 + LambertW[-x])/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%t a[n_] := a[n] = n^(n - 2) + (1/n) Sum[Binomial[n, k] (n - k)^(n - k - 2) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]

%Y Cf. A000272, A001858, A001865, A057817, A133297, A218688, A323673, A332236.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Feb 07 2020