A384165 a(n) = Product_{k=0..n-1} (3*n+2*k).

1, 3, 48, 1287, 48384, 2340135, 138378240, 9672183675, 780151357440, 71322093677835, 7287813911347200, 823100991923184975, 101819334240239616000, 13690816766440373134575, 1988199345147516813312000, 310120801435080997013527875, 51709528644340997758648320000

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = 2^n * RisingFactorial(3*n/2,n).

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

a(n) = (3/5) * 2^n * n! * binomial(5*n/2,n) for n > 0.

a(n) ~ 5^((5*n-1)/2) * n^n / (3^((3*n-1)/2) * exp(n)). - Amiram Eldar, Dec 08 2025

Mathematica

a[n_]:=Product[(3*n+2*k), {k, 0, n-1}]; Table[a[n], {n, 0, 15}] (* Vincenzo Librandi, May 22 2025 *)
a[n_] := (3/5) * 2^n * n! * Binomial[5*n/2, n]; a[0] = 1; Array[a, 18, 0] (* Amiram Eldar, Dec 08 2025 *)

Prog

(PARI) a(n) = prod(k=0, n-1, 3*n+2*k);
(SageMath)
def a(n): return 2^n*rising_factorial(3*n/2, n)
(Python)
from math import prod
def A384165(n): return prod(3*n+i for i in range(0, n<<1, 2)) # Chai Wah Wu, May 21 2025
(Magma) [1] cat  [&*[(3*n + 2*k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025

Crossrefs

Cf. A384164, A384166.

Sequence in context: A320668 A319732 A351424 * A199012 A383862 A304208

Adjacent sequences: A384162 A384163 A384164 * A384166 A384167 A384168

Keyword

nonn,easy

Author

Seiichi Manyama, May 21 2025

Status

approved