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%I #7 Apr 26 2021 12:19:27
%S 1,5,51,782,15992,408814,12541010,448834728,18358297416,844755218400,
%T 43190363326992,2429044756967520,149029669269441456,
%U 9905401062535389072,709016063545908259248,54375505616232613595904,4448148376192382963462400,386619861956492109750650496,35580548688887294090357622912
%N a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).
%F E.g.f.: 1 / (1 - 4*x + log(1 - x)).
%F a(n) ~ n! / ((4/c + 3 - c) * (1 - c/4)^n), where c = LambertW(4*exp(3)) = 3.2176447220005493578369738... - _Vaclav Kotesovec_, Apr 26 2021
%t a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[1/(1 - 4 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
%Y Cf. A007840, A052820, A053487, A343685, A343686.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 26 2021