A068943 a(n) = f(n, n, n), where f is the generalized super falling factorial (see comments for definition.).

1, 2, 24, 331776, 2524286414780230533120, 18356962141505758798331790171539976807981714702571497465907439808868887035904000000

Offset: 1

Author

Francois Jooste (phukraut(AT)hotmail.com), Mar 09 2002

Keywords

• nonn

Comments

f(x, p, r) = Product_{m = 1..p} (x-m+1)^binomial(m+r-2, m-1), for x > 0, x >= p >= 0, r > 0. f is a generalization of both the multi-level factorial A066121(n, k) and the falling factorial A068424(x, n). f(n, n, 1) = n! and f(n, n, 2) = the superfactorial A000178(n). In general f(n, n, r) = A066121(n+r, r+1). f(x, p, 1) = A068424(x, p) and f(x, p, r+1) = Product_{i = 0..p-1} f(x-i, p-i, r).

a(8) has 1213 digits. - Michael S. Branicky, Apr 09 2023

Examples

			a(3) = 24 since (4-1)^binomial(1+3-2,1-1) * (4-2)^binomial(2+3-2,2-1) * (4-3)^binomial(3+3-2,3-1) = 3^1 * 2^3 * 1 = 24.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..7

Programs

Maple

f := (x,p,r)->`if`(r<>0,`if`(p>0,product((x-m+1)^binomial(m+r-2,m-1),m=1..p),1),x); f(n,n,n);

PARI

a(n)=prod(m=1,n, (n-m+1)^binomial(m+n-2,m-1)) \\ Charles R Greathouse IV, Oct 30 2021

Python

from math import comb, prod
def a(n): return prod((n-m+1)**comb(m+n-2, m-1) for m in range(1, n+1))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Apr 09 2023

Formula

a(n) = Product_{m = 1..n} (n-m+1)^binomial(m+n-2, m-1).

Extensions

Edited by David Wasserman, Mar 14 2003