A369268 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^3 ).

1, 1, 2, 8, 29, 105, 414, 1695, 7046, 29853, 128644, 561262, 2474142, 11006108, 49343508, 222715440, 1011217425, 4615519083, 21165513228, 97467424198, 450541090701, 2089777230606, 9723511785608, 45371996501895, 212271904284993, 995513843930049, 4679212044797252

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. A071969, A369266.

Cf. A369269, A369270.

Programs

PARI

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

PARI

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

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