A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)^2), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

1, 2, 19, 514, 22621, 1369546, 105616639, 9901346554, 1093292035609, 138977379784882, 19990424969236171, 3209995501651871890, 569216406245186726965, 110476637766622355475898, 23294266811686640511534199, 5302371488162151660366545866, 1295920217231693678343467474353

Offset: 0

Author

Seiichi Manyama, Mar 12 2025

Keywords

• nonn

Crossrefs

Cf. A381987, A381988.

Cf. A002293, A002295, A382001.

Programs

PARI

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A382001. F(x) = B(x*A(x)^2) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002295(k)/(n-k)!.