A380643 Expansion of e.g.f. exp(x*G(3*x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 19, 865, 63289, 6402421, 827951491, 130454402149, 24246255965905, 5193341198368489, 1259626725043888051, 341256073037890028041, 102138911537774675080969, 33470594059698797005874845, 11918817613356955871120346979, 4582850483720783516657005897741

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

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

E.g.f. A(x) satisfies x = log(A(x)) * (1 - 3*log(A(x)))^3.

a(n) = 3^(n-1)*U(1-n, 2-4*n, 1/3), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025

E.g.f.: exp( Series_Reversion( x*(1-3*x)^3 ) ). - Seiichi Manyama, Mar 16 2025

Prog

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

Crossrefs

Cf. A002293, A380637.

Cf. A080893, A380640.

Sequence in context: A135562 A139194 A176601 * A334565 A183473 A217826

Adjacent sequences: A380640 A380641 A380642 * A380644 A380645 A380646

Keyword

nonn,changed

Author

Seiichi Manyama, Jan 29 2025

Status

approved