A369270 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^3)^3 ).

1, 3, 15, 94, 657, 4902, 38233, 307953, 2541831, 21386810, 182754162, 1581699162, 13836248406, 122139271098, 1086638457429, 9733419373534, 87707244737511, 794505072627735, 7231017033165776, 66089527981542462, 606340568510978940, 5582088822346925210

Offset: 0

Author

Seiichi Manyama, Jan 18 2024

Keywords

• nonn

Links

• P. Bala, Fractional iteration of a series inversion operator
• Index entries for reversions of series

Crossrefs

Cf. A369268, A369269.

Cf. A365843, A369264.

Cf. A369232.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+x^3)^3)/x)

PARI

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(4*n-3*k+2,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024