A381773 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^3 ) )^(1/3), where C(x) is the g.f. of A000108.

1, 2, 15, 157, 1913, 25427, 357546, 5229980, 78765793, 1213181593, 19021747383, 302595975502, 4871780511910, 79232327379407, 1299767617080662, 21481625997258747, 357350097625089497, 5978708468143961925, 100537111802285439375, 1698302173359384479307

Offset: 0

Author

Seiichi Manyama, Mar 07 2025

Keywords

• nonn

Crossrefs

Cf. A054727, A060941, A381772, A381774, A381775.

Cf. A212071, A381782, A381783.

Cf. A234461, A381779, A381780, A381786.

Cf. A000108.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^3)/x)^(1/3))

Formula

G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * C(x*A(x)^3).

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