A384262 a(n) = Product_{k=0..n-1} (3*n+k-2).

1, 1, 20, 504, 17160, 742560, 39070080, 2422728000, 173059286400, 13995229248000, 1264020397516800, 126115611484262400, 13776096431889792000, 1635195634511530291200, 209574860127295703040000, 28844656968251942737920000, 4243193364951971128258560000, 664387519844376163893657600000

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = RisingFactorial(3*n-2,n).

a(n) = n! * [x^n] 1/(1 - x)^(3*n-2).

a(n) = n! * binomial(4*n-3,n).

D-finite with recurrence 3*(3*n-4)*(3*n-5)*a(n) - 8*(4*n-5)*(4*n-3)*(2*n-3)*a(n-1) = 0. - R. J. Mathar, May 26 2025

a(n) ~ 2^(8*n-5) * n^n / (3^(3*n-5/2) * exp(n)). - Amiram Eldar, Dec 08 2025

Mathematica

a[n_] := n! * Binomial[4*n-3, n]; Array[a, 18, 0] (* Amiram Eldar, Dec 08 2025 *)

Prog

(PARI) a(n) = prod(k=0, n-1, 3*n+k-2);
(Python)
from sympy import rf
def a(n): return rf(3*n-2, n)
(SageMath)
def a(n): return rising_factorial(3*n-2, n)

Crossrefs

Cf. A061924, A384164, A384263.

Cf. A335647.

Sequence in context: A118552 A270578 A255492 * A092087 A008270 A130186

Adjacent sequences: A384259 A384260 A384261 * A384263 A384264 A384265

Keyword

nonn,easy

Author

Seiichi Manyama, May 23 2025

Status

approved