A334240 a(n) = exp(-n) * Sum_{k>=0} (k + 1)^n * n^k / k!.

1, 2, 11, 103, 1357, 23031, 478207, 11741094, 332734521, 10689163687, 383851610331, 15236978883127, 662491755803269, 31311446539427926, 1598351161031967063, 87638233726766111731, 5136809177699534717169, 320521818480481139673919, 21212211430440994022892019

Offset: 0

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..368
• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A242817, A299824, A334162, A334241, A334242, A334243.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[x + n (Exp[x] - 1)], {x, 0, n}], {n, 0, 18}]
Table[Sum[Binomial[n, k] BellB[k, n], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (n*x/(1 - x))^k / Product_{j=1..k} (1 - j*x/(1 - x)).

a(n) = n! * [x^n] exp(x + n*(exp(x) - 1)).

a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(n).

a(n) ~ exp((1/LambertW(1) - 2)*n) * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jun 08 2020