login
Expansion of e.g.f. exp(log(1 + x)^3).
2

%I #16 Feb 24 2023 11:17:57

%S 1,0,0,6,-36,210,-990,2184,37128,-863736,13020480,-168384744,

%T 1940801544,-18825129648,107706637584,1386022834944,-73429347222720,

%U 2034345021802560,-46869707752067520,976421492688165120,-18675350766042871680,319467427583225518080

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

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

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

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^3)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^2))

%o (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;

%o (PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/k!);

%Y Cf. A009199, A354232.

%Y Cf. A353344, A354136, A354229.

%K sign

%O 0,4

%A _Seiichi Manyama_, May 20 2022