A334242 a(n) = exp(-n) * Sum_{k>=0} (k + n)^n * n^k / k!.

1, 2, 18, 273, 5812, 159255, 5336322, 211385076, 9663571400, 500742188415, 29002424377110, 1856728690107027, 130194428384173116, 9923500366931329282, 816909605562423271178, 72231668379957026776065, 6827368666949651984215824, 686970682778467688690704639

Offset: 0

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A134980, A242817, A334240, A334241, A334243.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[n (Exp[x] + x - 1)], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[k, n] n^(n - k), {k, 0, n}], {n, 1, 17}]]

Formula

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

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

a(n) ~ c * exp((r^2/(1-r) - 1)*n) * n^n / (1-r)^n, where r = A333761 = 0.59894186245845296434937... is the root of the equation LambertW(r) = 1-r and c = 0.897950293373062982395233981707095204244165706668136925178217032608352851... - Vaclav Kotesovec, Jun 09 2020