A369263 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

1, 2, 10, 54, 329, 2126, 14356, 100030, 713956, 5193064, 38354066, 286860714, 2168308302, 16537766036, 127114940840, 983657456878, 7657060437148, 59917814944376, 471062428422152, 3718952705982232, 29471640802526185, 234356062245289566, 1869405604537134116

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. A218045, A369208.

Cf. A369262, A369264.

Cf. A370245.

Programs

PARI

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

PARI

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

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