A380799 Expansion of e.g.f. ( (1/x) * Series_Reversion( x * exp(-x * (1 + x)^2) / (1 + x)^3 ) )^2.

1, 8, 130, 3344, 119808, 5547112, 316221904, 21462652080, 1692342355840, 152162079949448, 15373938883590144, 1725108070356807952, 212915967853642332160, 28672289555680558679400, 4184239024352928346482688, 657856889310116430352244528, 110868321594997440513876197376

Offset: 0

Author

Seiichi Manyama, Feb 04 2025

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A380800.

Programs

PARI

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

Formula

E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * (1 + x*A(x)^(1/2))^2 ) * (1 + x*A(x)^(1/2))^6.

a(n) = 2 * n! * Sum_{k=0..n} (n+2)^(k-1) * binomial(3*n+2*k+6,n-k)/k!.