cp's OEIS Frontend

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

%I A068943 #14 Apr 09 2023 11:49:50
%S A068943 1,2,24,331776,2524286414780230533120,
%T A068943 18356962141505758798331790171539976807981714702571497465907439808868887035904000000
%N A068943 a(n) = f(n, n, n), where f is the generalized super falling factorial (see comments for definition.).
%C A068943 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).
%C A068943 a(8) has 1213 digits. - _Michael S. Branicky_, Apr 09 2023
%H A068943 Michael S. Branicky, <a href="/A068943/b068943.txt">Table of n, a(n) for n = 1..7</a>
%F A068943 a(n) = Product_{m = 1..n} (n-m+1)^binomial(m+n-2, m-1).
%e A068943 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.
%p A068943 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);
%o A068943 (PARI) a(n)=prod(m=1,n, (n-m+1)^binomial(m+n-2,m-1)) \\ _Charles R Greathouse IV_, Oct 30 2021
%o A068943 (Python)
%o A068943 from math import comb, prod
%o A068943 def a(n): return prod((n-m+1)**comb(m+n-2, m-1) for m in range(1, n+1))
%o A068943 print([a(n) for n in range(1, 8)]) # _Michael S. Branicky_, Apr 09 2023
%K A068943 nonn
%O A068943 1,2
%A A068943 Francois Jooste (phukraut(AT)hotmail.com), Mar 09 2002
%E A068943 Edited by _David Wasserman_, Mar 14 2003