A381983 E.g.f. A(x) satisfies A(x) = exp(x) * C(x*A(x)^2), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

1, 2, 15, 280, 8365, 342566, 17839339, 1128217084, 83987669721, 7194842276842, 697216089189511, 75408952092397760, 9005278056681754885, 1176889697125038323662, 167076740069554538243427, 25603739419854491589361636, 4212587964283017439802066353, 740650326150658335888643004498

Offset: 0

Author

Seiichi Manyama, Mar 11 2025

Keywords

• nonn

Crossrefs

Cf. A349640, A381982.

Cf. A000108, A002293, A381997.

Programs

PARI

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(4*k+1, k)/((4*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A381997. F(x) = C(x*A(x)^2) = exp( 1/2 * Sum_{k>=1} binomial(2*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002293(k)/(n-k)!.