A066121 -id:A066121 - cp's OEIS frontend

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

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

nonn

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

			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.

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

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);

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

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

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

Edited by David Wasserman, Mar 14 2003

A066119 Multi-level primorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=p(n) and a(n,n)=2.

Original entry on oeis.org

2, 3, 2, 5, 6, 2, 7, 30, 12, 2, 11, 210, 360, 24, 2, 13, 2310, 75600, 8640, 48, 2, 17, 30030, 174636000, 653184000, 414720, 96, 2, 19, 510510, 5244319080000, 114069441024000000, 270888468480000, 39813120, 192, 2, 23, 9699690

Offset: 1

Henry Bottomley, Dec 05 2001

			a(4,3)=a(3,2)*a(3,3)=6*2=12. Rows start 2; 3,2; 5,6,2; 7,30,12,2; ...

Columns include A000040, A002110, A006939 and A066120. Right hand side includes A007395 and A007283. Cf. A066121.

cp's OEIS Frontend

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

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A066119 Multi-level primorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=p(n) and a(n,n)=2.

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