A343673 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

1, 4, 33, 409, 6759, 139621, 3460989, 100091335, 3308146179, 123005753041, 5081871122073, 230948185830187, 11449697796242319, 614944043618257237, 35568197580789653685, 2204201734650777596863, 145703352769994600516187, 10233323176300508748808921, 761004837938469796089586257

Offset: 0

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

• nonn

Crossrefs

Cf. A000670, A005494, A006155, A032032, A343672, A343674.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 3 x - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (2 - 3*x - exp(x)).

a(n) ~ n! * 3^n / ((1 + LambertW(exp(2/3)/3)) * (2 - 3*LambertW(exp(2/3)/3))^(n+1)). - Vaclav Kotesovec, Jun 20 2022