A369296 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x^3)^2 ).

1, 1, 2, 7, 24, 84, 315, 1225, 4859, 19646, 80739, 336050, 1413587, 6000777, 25674462, 110598855, 479286932, 2088036939, 9139604421, 40174594432, 177267942918, 784889441217, 3486198469890, 15529021825140, 69355660644738, 310509670642611, 1393296782758244

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. A063030, A369299.

Cf. A370274.

Programs

PARI

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

PARI

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

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