A347004 Expansion of e.g.f. exp( -log(1 - x)^5 / 5! ).

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269451, 3423860, 46238280, 664233856, 10143487354, 164423078456, 2823768543960, 51272283444264, 982177492263750, 19807082824819374, 419629806223448346, 9320808413229618816, 216645165604679499072, 5259724543984442886486

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A000482, A327506, A346924, A347001, A347002, A347003.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Exp[-Log[1 - x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

PARI

a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/(120^k*k!)); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/(120^k * k!). - Seiichi Manyama, May 06 2022