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

Author

Seiichi Manyama, May 21 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A384164, A384165.

Cf. A000407, A352601.

Cf. A061924, A124320.

Programs

Magma

[1] cat  [&*[(3*n + k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025

Mathematica

a[n_]:=Product[(3*n+k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)

PARI

a(n) = prod(k=0, n-1, 3*n+k);

Python

from sympy import rf
def A384164(n): return rf(3*n,n) # Chai Wah Wu, May 21 2025

Sage

def a(n): return rising_factorial(3*n, n)

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.