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A354231
Expansion of e.g.f. exp(log(1 + x)^3).
2
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
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
KEYWORD
sign
AUTHOR
Seiichi Manyama, May 20 2022
STATUS
approved