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a(n) = exp(-n) * Sum_{k>=0} (k + n)^n * n^k / k!.
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%I #8 Jun 09 2020 07:17:31

%S 1,2,18,273,5812,159255,5336322,211385076,9663571400,500742188415,

%T 29002424377110,1856728690107027,130194428384173116,

%U 9923500366931329282,816909605562423271178,72231668379957026776065,6827368666949651984215824,686970682778467688690704639

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>

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

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

%F 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

%t Table[n! SeriesCoefficient[Exp[n (Exp[x] + x - 1)], {x, 0, n}], {n, 0, 17}]

%t Join[{1}, Table[Sum[Binomial[n, k] BellB[k, n] n^(n - k), {k, 0, n}], {n, 1, 17}]]

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 19 2020