A334193 a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (n*k + 1)^n / ((-n)^k * k!).

1, 0, -2, -9, -16, 625, 21384, 571438, 13471744, 188661555, -9794500000, -1476328587789, -134710712340480, -10664210861777200, -744650964057237888, -37832162051689453125, 831929248561267474432, 725944099523076464203157, 167435684777981700601449984

Offset: 0

Author

Ilya Gutkovskiy, Apr 18 2020

Keywords

• sign

Crossrefs

Cf. A318183, A334162, A334190, A334191, A334192.

Programs

Mathematica

Table[SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[x + (1 - Exp[n x])/n], {x, 0, n}], {n, 1, 18}]]
Join[{1}, Table[Sum[Binomial[n, k]*n^k*BellB[k, -1/n], {k, 0, n}], {n, 1, 18}]] (* Vaclav Kotesovec, Apr 18 2020 *)

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - n*j*x/(1 - x)).

a(n) = n! * [x^n] exp(x + (1 - exp(n*x)) / n), for n > 0.

a(n) = A334192(n,n).