A347004 - OEIS

%I #11 May 06 2022 20:34:05

%S 1,0,0,0,0,1,15,175,1960,22449,269451,3423860,46238280,664233856,

%T 10143487354,164423078456,2823768543960,51272283444264,

%U 982177492263750,19807082824819374,419629806223448346,9320808413229618816,216645165604679499072,5259724543984442886486

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

%H Seiichi Manyama, <a href="/A347004/b347004.txt">Table of n, a(n) for n = 0..448</a>

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

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

%t nmax = 23; CoefficientList[Series[Exp[-Log[1 - x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]!

%t 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}]

%o (PARI) a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/(120^k*k!)); \\ _Seiichi Manyama_, May 06 2022

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

%K nonn

%O 0,7

%A _Ilya Gutkovskiy_, Aug 10 2021