A381819 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, 16, 177, 2271, 31731, 468614, 7195295, 113712012, 1837457589, 30220139048, 504212998955, 8513461623355, 145197727340337, 2497695979786842, 43285207907364178, 755005614380697735, 13244500528948104210, 233515959911770430972, 4135792046643993604967

Offset: 0

Author

Seiichi Manyama, Mar 07 2025

Keywords

• nonn

Crossrefs

Cf. A381817, A381818, A381820.

Cf. A000108, A381773.

Programs

PARI

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

Formula

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

a(n) = Sum_{k=0..n} binomial(3*n+2*k+1,k) * binomial(4*n-k,n-k)/(3*n+2*k+1).