A216919 The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals.

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, 15, 40, 3, 6, 1, 1, 1, 40320, 105, 40, 15, 24, 1, 2, 1, 1, 362880, 105, 280, 15, 24, 1, 6, 1, 1, 1, 3628800, 945, 2240, 105, 144, 5, 24, 3, 2, 1, 1, 39916800, 945

Offset: 1

Author

Peter Luschny, Oct 01 2012

Keywords

• nonn
• tabl

Comments

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.

Examples

			[n\N][0, 1, 2, 3,  4,   5,   6,    7,     8,      9,     10]
------------------------------------------------------------
[  1] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A000142]
[  2] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945 [A055634, A133221]
[  3] 1, 1, 2, 2,  8,  40,  40,  280,  2240,   2240,   22400 [A232980]
[  4] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945
[  5] 1, 1, 2, 6, 24,  24, 144, 1008,  8064,  72576,   72576 [A232981]
[  6] 1, 1, 1, 1,  1,   5,   5,   35,    35,     35,      35 [A232982]
[  7] 1, 1, 2, 6, 24, 120, 720,  720,  5760,  51840,  518400 [A232983]
[  8] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945
[  9] 1, 1, 2, 2,  8,  40,  40,  280,  2240,   2240,   22400
[ 10] 1, 1, 1, 3,  3,   3,   3,   21,    21,    189,     189 [A232984]
[ 11] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A232985]
[ 12] 1, 1, 1, 1,  1,   5,   5,   35,    35,     35,      35
[ 13] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, flattened
• J. B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008).
• J. B. Cosgrave, K. Dilcher, An Introduction to Gauss Factorials, The American Mathematical Monthly, Vol. 118, No. 9 (2011), 812-829.
• K. Dilcher, Gauss Factorials: Properties and Applications. Video by the Irmacs Centre, May 18, 2011.

Crossrefs

A000142(n) = n! = Gauss_factorial(n, 1).

A001147(n) = Gauss_factorial(2*n, 2).

A055634(n) = Gauss_factorial(n, 2)*(-1)^n.

A001783(n) = Gauss_factorial(n, n).

A124441(n) = Gauss_factorial(floor(n/2), n).

A124442(n) = Gauss_factorial(n, n)/Gauss_factorial(floor(n/2), n).

A066570(n) = Gauss_factorial(n, 1)/Gauss_factorial(n, n).

Cf. A133221, A232980, A232981, A232982, A232983, A232984, A232985.

Programs

Maple

A:= (n, N)-> mul(`if`(igcd(j, n)=1, j, 1), j=1..N):
seq(seq(A(n, d-n), n=1..d), d=1..12);  # Alois P. Heinz, Oct 03 2012

Mathematica

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 *)

PARI

T(m,n)=prod(k=2, m, if(gcd(k,n)==1, k, 1))
for(s=1,10,for(n=1,s,print1(T(s-n,n)", "))) \\ Charles R Greathouse IV, Oct 01 2012

Sage

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
for n in (1..13): [Gauss_factorial(N, n) for N in (0..10)]

Formula

N_n! = product_{1<=j<=N, GCD(j,n)=1} j.