A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217

Offset: 0

Author

Seiichi Manyama, Jan 26 2025

Keywords

• nonn

Crossrefs

Cf. A380513, A380514, A380516.

Cf. A080893, A380511.

Cf. A091695, A250917, A380512.

Cf. A002293, A006632, A370057, A382059.

Programs

PARI

a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

Formula

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.

a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025

E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025