A354232 - OEIS

%I #9 May 20 2022 08:50:39

%S 1,0,0,0,0,120,-1800,21000,-235200,2693880,-30504600,310239600,

%T -2026767600,-22324267680,1480359360480,-48314853350400,

%U 1332965821824000,-34178451017685120,837433109548661760,-19671723873906894720,436228097513559408000

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

%F E.g.f.: (1 + x)^(log(1 + x)^4).

%F a(0) = 1; a(n) = 120 * 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)/k!.

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^5)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^4))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 1)*v[i-j+1])); v;

%o (PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/k!);

%Y Cf. A009199, A354231.

%Y Cf. A354137, A354230.

%K sign

%O 0,6

%A _Seiichi Manyama_, May 20 2022