A354231 Expansion of e.g.f. exp(log(1 + x)^3).

1, 0, 0, 6, -36, 210, -990, 2184, 37128, -863736, 13020480, -168384744, 1940801544, -18825129648, 107706637584, 1386022834944, -73429347222720, 2034345021802560, -46869707752067520, 976421492688165120, -18675350766042871680, 319467427583225518080

Offset

0,4

Links

Table of n, a(n) for n=0..21.

Formula

E.g.f.: (1 + x)^(log(1 + x)^2).

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,3) * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/k!.

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^3)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^2))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 1)*v[i-j+1])); v;
(PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/k!);

Crossrefs

Cf. A009199, A354232.

Cf. A353344, A354136, A354229.

Sequence in context: A111989 A064238 A354229 * A014336 A268941 A269637

Adjacent sequences: A354228 A354229 A354230 * A354232 A354233 A354234

Keyword

sign

Author

Seiichi Manyama, May 20 2022

Status

approved