A382032 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, 1, 5, 55, 937, 21741, 639841, 22839139, 958882289, 46304377849, 2528571710881, 154076164781991, 10364272238514217, 762867688235619877, 60989719558159065857, 5263030218009265964011, 487578723768665716788961, 48266847740986728218648433, 5084697384633390178057209793

Offset: 0

Author

Seiichi Manyama, Mar 12 2025

Keywords

• nonn

Crossrefs

Cf. A161630, A382033, A382034.

Cf. A000108, A377553, A382036.

Programs

PARI

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

Formula

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

Let F(x) be the e.g.f. of A377553. F(x) = log(A(x))/x = C(x*A(x))^2.

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