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%I #6 Aug 13 2020 22:43:10
%S 1,1,3,12,72,534,4818,50532,606408,8182656,122712912,2024328096,
%T 36432644400,710346495312,14915647605168,335567743462944,
%U 8052843408926976,205328108580310656,5543345188496499840,157970863597032124416,4738694884696030305024
%N E.g.f.: 1 / (1 + x^3/3 + log(1 - x)).
%F a(0) = a(1) = 1; a(n) = n * (a(n-1) + (n-1) * a(n-2) / 2) + Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).
%t nmax = 20; CoefficientList[Series[1/(1 + x^3/3 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = a[1] = 1; a[n_] := a[n] = n (a[n - 1] + (n - 1) a[n - 2]/2) + Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 20}]
%Y Cf. A000090, A007840, A226226, A337060.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 13 2020