A355379 Expansion of e.g.f. exp(exp(3*x) + exp(x) - 2).

1, 4, 26, 212, 2046, 22588, 278942, 3792916, 56128254, 895795692, 15307847614, 278435732484, 5364073445278, 108994074306268, 2327475127169182, 52069279762495220, 1217024509006768574, 29647115491635327180, 751085909757123127294, 19750410883486281805028

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..466

Formula

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

a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022

a(n) ~ exp(exp(3*z) + exp(z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) + (1 + z)*exp(z)), where z = LambertW(n)/3 - 1/(1 + 3/LambertW(n) + 9 * n^(2/3) * (1 + LambertW(n)) / LambertW(n)^(5/3)). - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) + (n/LambertW(n))^(1/3) - n - 2) / sqrt(1 + LambertW(n)). - Vaclav Kotesovec, Jul 10 2022

Mathematica

nmax = 20; CoefficientList[Series[Exp[Exp[3*x] + Exp[x] - 2], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n, k] * 3^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) + exp(x) - 2))) \\ Michel Marcus, Jun 30 2022

Crossrefs

Cf. A143405, A355291, A355378.

Sequence in context: A363363 A239295 A274735 * A349719 A283828 A306028

Adjacent sequences: A355376 A355377 A355378 * A355380 A355381 A355382

Keyword

nonn

Author

Vaclav Kotesovec, Jun 30 2022

Status

approved