A384164 a(n) = Product_{k=0..n-1} (3*n+k).

1, 3, 42, 990, 32760, 1395360, 72681840, 4475671200, 318073392000, 25622035084800, 2306992893004800, 229601607198163200, 25028504609870361600, 2965681982933429760000, 379534960108578193920000, 52170410224819317150720000, 7666009844358186506465280000, 1199151678674216896627654656000

Offset

0,2

Links

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

Formula

a(n) = RisingFactorial(3*n,n) = A124320(3*n,n) = n! * binomial(4*n-1,n).

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

a(n) = (3/4) * A061924(n) for n > 0.

a(n) ~ 2^(8*n-1) * n^n / (3^(3*n-1/2) * exp(n)). - Amiram Eldar, Dec 08 2025

Mathematica

a[n_]:=Product[(3*n+k), {k, 0, n-1}]; Table[a[n], {n, 0, 15}] (* Vincenzo Librandi, May 22 2025 *)
a[n_] := n! * Binomial[4*n-1, n]; Array[a, 18, 0] (* Amiram Eldar, Dec 08 2025 *)

Prog

(PARI) a(n) = prod(k=0, n-1, 3*n+k);
(SageMath)
def a(n): return rising_factorial(3*n, n)
(Python)
from sympy import rf
def A384164(n): return rf(3*n, n) # Chai Wah Wu, May 21 2025
(Magma) [1] cat  [&*[(3*n + k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025

Crossrefs

Cf. A384164, A384165.

Cf. A000407, A352601.

Cf. A061924, A124320.

Sequence in context: A366010 A206820 A157542 * A078601 A268621 A380097

Adjacent sequences: A384161 A384162 A384163 * A384165 A384166 A384167

Keyword

nonn,easy

Author

Seiichi Manyama, May 21 2025

Status

approved