A355111 Expansion of e.g.f. 3 / (4 - 3*x - exp(3*x)).

1, 2, 11, 93, 1041, 14541, 243747, 4767183, 106556373, 2679469065, 74864397015, 2300883358995, 77144051804409, 2802027511061325, 109604157405491691, 4593512301562215783, 205348466229473678301, 9753645833118762303249, 490530576727430107027839, 26040317900991310393061499

Offset: 0

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

• nonn

Crossrefs

Cf. A003575, A006155, A032033, A255927, A343673, A355110, A355112, A355113, A355114.

Programs

Mathematica

nmax = 19; CoefficientList[Series[3/(4 - 3 x - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 3^(k-1) * a(n-k).

a(n) ~ n! / ((1 + LambertW(exp(4))) * ((4 - LambertW(exp(4)))/3)^(n+1)). - Vaclav Kotesovec, Jun 19 2022