A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(x*A(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

1, 2, 11, 139, 2829, 78981, 2802163, 120667667, 6113752025, 356342305465, 23488872131871, 1727770084512495, 140302645206245701, 12466960491079733237, 1203253101643330233707, 125351056198801059896491, 14019427299278115378992049, 1675439381194882102492648305

Offset: 0

Author

Seiichi Manyama, Mar 11 2025

Keywords

• nonn

Crossrefs

Cf. A349640, A381983.

Cf. A000108, A001764, A161629, A364983.

Programs

PARI

a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(3*k+1, k)/((3*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A364983. F(x) = C(x*A(x)) = exp( 1/2 * Sum_{k>=1} binomial(2*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A001764(k)/(n-k)!.