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%I #35 Jul 11 2022 14:57:10
%S 1,1,1,2,1,1,6,1,1,1,24,3,2,1,1,120,3,2,1,1,1,720,15,8,3,2,1,1,5040,
%T 15,40,3,6,1,1,1,40320,105,40,15,24,1,2,1,1,362880,105,280,15,24,1,6,
%U 1,1,1,3628800,945,2240,105,144,5,24,3,2,1,1,39916800,945
%N The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals.
%C The term is due to Cosgrave & Dilcher. The Gauss factorial should not be confused with the q-factorial [n]_q! which is also called Gaussian factorial.
%H Alois P. Heinz, <a href="/A216919/b216919.txt">Antidiagonals n = 1..141, flattened</a>
%H J. B. Cosgrave, K. Dilcher, <a href="http://www.emis.de/journals/INTEGERS/papers/i39/i39.Abstract.html"> Extensions of the Gauss-Wilson Theorem</a>, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008).
%H J. B. Cosgrave, K. Dilcher, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.09.812">An Introduction to Gauss Factorials</a>, The American Mathematical Monthly, Vol. 118, No. 9 (2011), 812-829.
%H K. Dilcher, <a href="http://vimeo.com/25261314">Gauss Factorials: Properties and Applications</a>. Video by the Irmacs Centre, May 18, 2011.
%F N_n! = product_{1<=j<=N, GCD(j,n)=1} j.
%e [n\N][0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
%e ------------------------------------------------------------
%e [ 1] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A000142]
%e [ 2] 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945 [A055634, A133221]
%e [ 3] 1, 1, 2, 2, 8, 40, 40, 280, 2240, 2240, 22400 [A232980]
%e [ 4] 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945
%e [ 5] 1, 1, 2, 6, 24, 24, 144, 1008, 8064, 72576, 72576 [A232981]
%e [ 6] 1, 1, 1, 1, 1, 5, 5, 35, 35, 35, 35 [A232982]
%e [ 7] 1, 1, 2, 6, 24, 120, 720, 720, 5760, 51840, 518400 [A232983]
%e [ 8] 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945
%e [ 9] 1, 1, 2, 2, 8, 40, 40, 280, 2240, 2240, 22400
%e [ 10] 1, 1, 1, 3, 3, 3, 3, 21, 21, 189, 189 [A232984]
%e [ 11] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A232985]
%e [ 12] 1, 1, 1, 1, 1, 5, 5, 35, 35, 35, 35
%e [ 13] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800
%p A:= (n, N)-> mul(`if`(igcd(j, n)=1, j, 1), j=1..N):
%p seq(seq(A(n, d-n), n=1..d), d=1..12); # _Alois P. Heinz_, Oct 03 2012
%t GaussFactorial[m_, n_] := Product[ If[ GCD[j, n] == 1, j, 1], {j, 1, m}]; Table[ GaussFactorial[m - n, n], {m, 1, 12}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Mar 18 2013 *)
%o (Sage)
%o def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
%o for n in (1..13): [Gauss_factorial(N, n) for N in (0..10)]
%o (PARI) T(m,n)=prod(k=2, m, if(gcd(k,n)==1, k, 1))
%o for(s=1,10,for(n=1,s,print1(T(s-n,n)", "))) \\ _Charles R Greathouse IV_, Oct 01 2012
%Y A000142(n) = n! = Gauss_factorial(n, 1).
%Y A001147(n) = Gauss_factorial(2*n, 2).
%Y A055634(n) = Gauss_factorial(n, 2)*(-1)^n.
%Y A001783(n) = Gauss_factorial(n, n).
%Y A124441(n) = Gauss_factorial(floor(n/2), n).
%Y A124442(n) = Gauss_factorial(n, n)/Gauss_factorial(floor(n/2), n).
%Y A066570(n) = Gauss_factorial(n, 1)/Gauss_factorial(n, n).
%Y Cf. A133221, A232980, A232981, A232982, A232983, A232984, A232985.
%K nonn,tabl
%O 1,4
%A _Peter Luschny_, Oct 01 2012
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