A320082 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k/k!.

1, 1, 3, 5, -60, -186, 13832, -98862, -8631360, 352796880, 4245955032, -1185349047048, 48595690153920, 3201334718188320, -607575977909763840, 26489851912606455504, 4482546578798646251520, -958939334596403708474880, 50300999315063602037775360, 14223928928980522264922223360, -3933112779003946549567400925696

Offset: 0

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A048994, A108459, A128943, A232549.

Programs

Maple

1,seq(n!*coeff(series(add(log(1+k*x)^k/k!, k=1..100), x=0, 21), x, n), n=1..20); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 20; CoefficientList[Series[1 + Sum[Log[1 + k x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^n, {k, n}], {n, 20}]]

PARI

a(n) = sum(k=0, n, stirling(n,k)*k^n); \\ Altug Alkan, Oct 05 2018

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k^n.