A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + n*j*x).

1, 1, -1, 1, 25, -674, 15211, -331827, 5987745, 15901597, -13125035449, 1292056076070, -103145930581319, 7462324963409941, -464957409070517453, 16313974895147212801, 2059903411953959582849, -708700955022151333496910, 143215213612865558214820303, -24681846509158429152517973103

Offset: 0

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..282
• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A009235, A014182, A292866, A301419, A317996, A318179, A318180, A318181.

Programs

Mathematica

Table[SeriesCoefficient[Sum[x^k/Product[(1 + n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[(-n)^(n - k) StirlingS2[n, k], {k, n}], {n, 19}]]
Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* Peter Luschny, Aug 20 2018 *)

PARI

{a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 27 2019

Formula

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

a(n) = Sum_{k=0..n} (-n)^(n-k)*Stirling2(n,k).

a(n) = (-n)^n*BellPolynomial_n(-1/n) for n >= 1. - Peter Luschny, Aug 20 2018