%I #7 Aug 09 2021 03:15:03
%S 1,2,14,132,1676,26590,506202,11242952,285383240,8149464954,
%T 258575410190,9024809281972,343619185754748,14173557899208422,
%U 629600469603730562,29965010056866657600,1521221783964264806672,82053967063309770102130,4686301361507067542636694
%N E.g.f.: 1 / (1 - x * (2 + x) * exp(x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * (k + 1) * a(n-k).
%F a(n) ~ n! * (2 + r) / ((2 + 4*r + r^2) * r^n), where r = 0.31516782494427474715049117135360576083681438371... is the root of the equation exp(r) * r * (2 + r) = 1. - _Vaclav Kotesovec_, Aug 09 2021
%t nmax = 18; CoefficientList[Series[1/(1 - x (2 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k (k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%Y Cf. A002378, A006153, A308861, A308946, A336961.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Aug 09 2020