A365843 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ).

1, 6, 54, 578, 6810, 85278, 1113854, 15004746, 206955378, 2908113974, 41484917958, 599202514578, 8745727050762, 128790559374030, 1911191826600462, 28551332345784730, 429040549473424866, 6480799118506040934, 98349636147075506006, 1498732955394826784226

Offset: 0

Author

Seiichi Manyama, Sep 20 2023

Keywords

• nonn

Crossrefs

Column k=3 of A378238.

Cf. A007297, A365842, A365844, A365845.

Cf. A032349, A365622, A365847.

Cf. A144097.

Programs

PARI

a(n) = sum(k=0, n, binomial(3*n+k+2, k)*binomial(3*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*(n+1),n-k).

G.f.: B^3, where B is the g.f. of A144097.

a(n) ~ sqrt(8060 + 2651*sqrt(10)) * (223 + 70*sqrt(10))^n / (2 * sqrt(5*Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024