A355112 Expansion of e.g.f. 4 / (5 - 4*x - exp(4*x)).

1, 2, 12, 112, 1376, 21056, 386688, 8286720, 202958848, 5592199168, 171203895296, 5765504860160, 211811563929600, 8429932686999552, 361312700788375552, 16592261047219388416, 812749365813312487424, 42299637489384965537792, 2330989060564353634271232

Offset: 0

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

• nonn

Crossrefs

Cf. A003576, A006155, A094417, A326324, A343674, A355110, A355111, A355113, A355114.

Programs

Mathematica

nmax = 18; CoefficientList[Series[4/(5 - 4 x - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
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}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k).

a(n) ~ n! / ((1 + LambertW(exp(5))) * ((5 - LambertW(exp(5)))/4)^(n+1)). - Vaclav Kotesovec, Jun 19 2022