A369014 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^3 ).

1, 0, 0, 3, 3, 3, 36, 78, 129, 685, 2043, 4554, 17233, 57279, 153045, 509848, 1724739, 5117643, 16445555, 55165536, 173225715, 555899673, 1847495415, 5971507824, 19333284247, 63975307425, 209807070669, 685973054145, 2269660792842, 7501194321663, 24725092907853

Offset: 0

Author

Seiichi Manyama, Jan 11 2024

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for reversions of series

Crossrefs

Cf. A369012, A369013.

Cf. A054514, A369011.

Programs

PARI

my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x))^3)/x)

PARI

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

Formula

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