A307080 a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n*k + 1)^n/k!.

1, 0, -3, 19, 497, -1899, -489491, -15433676, 618450881, 120846851155, 7012261819901, -467816186167659, -175527285590430863, -20961845760818684812, 568194037748383908653, 898095630359015975379151, 220433074470274983356464897, 16144974747716546214909454181

Offset: 0

Author

Ilya Gutkovskiy, Jun 24 2019

Keywords

• sign

Crossrefs

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

Programs

Mathematica

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

Formula

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

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