A336183 a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.
1, 5, 23, 154, 1389, 15636, 211231, 3329264, 59969097, 1215233380, 27362096211, 677690995488, 18310602210445, 535964033279780, 16894811428737495, 570603293774677696, 20556251540382371217, 786832900592755991364, 31889277719673937849243, 1364231113649221829763200
Offset: 1
Keywords
Programs
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Mathematica
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}] nmax = 20; CoefficientList[Series[-Log[1 - Exp[x] x (1 + x)], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Formula
E.g.f.: -log(1 - exp(x) * x * (1 + x)).
E.g.f.: -log(1 - Sum_{k>=1} k^2 * x^k / k!).
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