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%I #6 Jun 20 2022 03:22:41
%S 1,5,51,781,15947,407021,12466251,445452813,18191122219,835737327661,
%T 42661645147403,2395510523568845,146739531459316587,
%U 9737742346694258157,695911661109898805323,53286006304099668950413,4352120920347139791200171,377674509364714706139413933,34702277449656625185428239755
%N a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).
%F E.g.f.: 1 / (2 * (1 - 2*x) - exp(x)).
%F a(n) ~ n! * 2^(n-1) / ((1 + LambertW(exp(1/2)/4)) * (1 - 2*LambertW(exp(1/2)/4))^(n+1)). - _Vaclav Kotesovec_, Jun 20 2022
%t a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[1/(2 (1 - 2 x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%Y Cf. A000670, A006155, A032032, A045379, A343672, A343673.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 25 2021