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A307066 a(n) = exp(-1) * Sum_{k>=0} (n*k + 1)^n/k!.

%I A307066 #45 Jan 25 2024 01:06:27
%S A307066 1,2,13,199,5329,216151,12211597,909102342,85761187393,9957171535975,
%T A307066 1390946372509101,229587693339867567,44117901231194922193,
%U A307066 9748599124579281064294,2451233017637221706477037,695088863051920283838281851,220558203335628758134165860609
%N A307066 a(n) = exp(-1) * Sum_{k>=0} (n*k + 1)^n/k!.
%H A307066 G. C. Greubel, <a href="/A307066/b307066.txt">Table of n, a(n) for n = 0..250</a>
%F A307066 a(n) = n! * [x^n] exp(exp(n*x) + x - 1).
%F A307066 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * Bell(k).
%t A307066 Table[Exp[-1] Sum[(n k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 16}]
%t A307066 Table[n! SeriesCoefficient[Exp[Exp[n x] + x - 1], {x, 0, n}], {n, 0, 16}]
%t A307066 Join[{1}, Table[Sum[Binomial[n, k] n^k BellB[k], {k, 0, n}], {n, 1, 16}]]
%o A307066 (Magma)
%o A307066 A307066:= func< n | (&+[Binomial(n,k)*n^k*Bell(k): k in [0..n]]) >;
%o A307066 [A307066(n): n in [0..31]]; // _G. C. Greubel_, Jan 24 2024
%o A307066 (SageMath)
%o A307066 def A307066(n): return sum(binomial(n,k)*n^k*bell_number(k) for k in range(n+1))
%o A307066 [A307066(n) for n in range(31)] # _G. C. Greubel_, Jan 24 2024
%Y A307066 Cf. A000110, A126390, A134980, A284859, A285064, A292914, A307080.
%K A307066 nonn
%O A307066 0,2
%A A307066 _Ilya Gutkovskiy_, Jun 24 2019