A354232 Expansion of e.g.f. exp(log(1 + x)^5).

1, 0, 0, 0, 0, 120, -1800, 21000, -235200, 2693880, -30504600, 310239600, -2026767600, -22324267680, 1480359360480, -48314853350400, 1332965821824000, -34178451017685120, 837433109548661760, -19671723873906894720, 436228097513559408000

Offset

0,6

Links

Table of n, a(n) for n=0..20.

Formula

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

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

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^5)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^4))
(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;
(PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/k!);

Crossrefs

Cf. A009199, A354231.

Cf. A354137, A354230.

Sequence in context: A056270 A001118 A354230 * A052767 A353404 A353200

Adjacent sequences: A354229 A354230 A354231 * A354233 A354234 A354235

Keyword

sign

Author

Seiichi Manyama, May 20 2022

Status

approved