A381831 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^3 ) )^(1/3).

1, 2, 14, 133, 1456, 17306, 217066, 2827896, 37895130, 519000037, 7232429952, 102220846756, 1461817707558, 21112968248198, 307527937374182, 4512344039147420, 66634574697351360, 989569163283434676, 14769533757869187052, 221426909287107012800, 3333042591222552282784, 50353576994047154278451

Offset: 0

Author

Seiichi Manyama, Mar 08 2025

Keywords

• nonn

Crossrefs

Cf. A364592, A381818, A381830.

Cf. A129442, A381828.

Cf. A000108.

Programs

PARI

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

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.

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