A369267 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^2 ).

1, 2, 7, 32, 163, 884, 5009, 29310, 175750, 1074264, 6668825, 41929970, 266464579, 1708829584, 11044663663, 71871779008, 470495357634, 3096311833496, 20472771422946, 135937759368388, 906056228361095, 6059922934991008, 40657629626645463

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. A369265, A369269.

Cf. A369266, A370249.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3)^2)/x)

PARI

a(n, s=3, t=2, u=2) = 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(2*n+2,k) * binomial(3*n-3*k+1,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024