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%I #8 Jul 11 2020 07:39:39
%S 1,5,23,154,1389,15636,211231,3329264,59969097,1215233380,27362096211,
%T 677690995488,18310602210445,535964033279780,16894811428737495,
%U 570603293774677696,20556251540382371217,786832900592755991364,31889277719673937849243,1364231113649221829763200
%N a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.
%F E.g.f.: -log(1 - exp(x) * x * (1 + x)).
%F E.g.f.: -log(1 - Sum_{k>=1} k^2 * x^k / k!).
%F a(n) ~ (n-1)! / r^n, where r = A201941 = 0.444130228823966590585466329490984667... is the root of the equation exp(r)*r*(1+r) = 1. - _Vaclav Kotesovec_, Jul 11 2020
%t a[n_] := a[n] = n^2 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
%t nmax = 20; CoefficientList[Series[-Log[1 - Exp[x] x (1 + x)], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%Y Cf. A000290, A033462, A033464, A305990, A308861, A336184.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Jul 10 2020