%I #7 Jun 21 2022 12:46:24
%S 1,2,12,112,1376,21056,386688,8286720,202958848,5592199168,
%T 171203895296,5765504860160,211811563929600,8429932686999552,
%U 361312700788375552,16592261047219388416,812749365813312487424,42299637489384965537792,2330989060564353634271232
%N Expansion of e.g.f. 4 / (5 - 4*x - exp(4*x)).
%F a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k).
%F a(n) ~ n! / ((1 + LambertW(exp(5))) * ((5 - LambertW(exp(5)))/4)^(n+1)). - _Vaclav Kotesovec_, Jun 19 2022
%t nmax = 18; CoefficientList[Series[4/(5 - 4 x - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%Y Cf. A003576, A006155, A094417, A326324, A343674, A355110, A355111, A355113, A355114.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 19 2022