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a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n*k + 1)^n/k!.
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%I #26 Jun 24 2019 20:09:50

%S 1,0,-3,19,497,-1899,-489491,-15433676,618450881,120846851155,

%T 7012261819901,-467816186167659,-175527285590430863,

%U -20961845760818684812,568194037748383908653,898095630359015975379151,220433074470274983356464897,16144974747716546214909454181

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

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

%F a(n) = Sum_{k=0..n} binomial(n,k) * n^k * A000587(k).

%t Table[Exp[1] Sum[(-1)^k (n k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 17}]

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

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

%Y Cf. A000587, A284860, A285065, A293037, A298373, A307066, A308645.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jun 24 2019