A370057 a(n) = 3*(4*n+2)!/(3*n+3)!.

1, 3, 30, 546, 14688, 526680, 23680800, 1282554000, 81339793920, 5915366392320, 485415660038400, 44376781223174400, 4473125162795520000, 492902545595556096000, 58949616073242166272000, 7605168496387089788160000, 1052810955815818170875904000

Offset: 0

Author

Seiichi Manyama, Feb 08 2024

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..327

Crossrefs

Cf. A365340, A370056, A370058.

Cf. A000142, A006632.

Programs

Mathematica

Table[(3(4n+2)!)/(3n+3)!,{n,0,20}] (* Harvey P. Dale, Feb 15 2025 *)

PARI

a(n) = 3*(4*n+2)!/(3*n+3)!;

Formula

E.g.f.: exp( 3/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).

a(n) = A000142(n) * A006632(n+1).

D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 22 2024

From Seiichi Manyama, Aug 31 2024: (Start)

E.g.f. satisfies A(x) = 1/(1 - x*A(x))^3.

a(n) = 3 * Sum_{k=0..n} (3*n+3)^(k-1) * |Stirling1(n,k)|. (End)