A397838 - OEIS

%I #6 Jul 13 2026 02:43:19

%S 1,1,2,12,144,5760,259200,58060800,9144576000,5618427494400,

%T 4147979673600000,8009630236016640000,7588811772444672000000,

%U 87964963104029147136000000,287964353346502458802176000000,2259278037969787970377482240000000,30746913887983889201171005440000000000

%N Antidiagonal products of the multifactorial array.

%C It appears that the only squares in this sequence are a(0) = a(1) = 1, a(4) = 144, a(18) = 14671475874842562153331362760375664640000000000.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.

%F a(n) = Product_{k=1..n+1} A(n+1-k, k), where A denotes one of the arrays A114423 and A129116.

%F a(n) = Product_{k=1..n} Product_{m=1..floor(n/k)} (n + 1 - m*k).

%t A[0, k_]:=1; A[n_, k_]:=A[n, k]=If[1 <= n <= k, n, n*A[n - k, k]]; a[n_]:=Product[A[n+1-k,k],{k,n+1}]; Array[a,17,0]

%t (* Alternative: *)

%t a[n_]:=Product[Product[n+1-m*k,{m,Floor[n/k]}],{k,n}]; Array[a,17,0]

%Y Cf. A000290, A010766, A114423, A129116.

%K nonn,new

%O 0,3

%A _Stefano Spezia_, Jul 12 2026