A216919 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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.

LINKS

Alois P. Heinz, Antidiagonals n = 1..141, flattened

J. B. Cosgrave and K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008).

J. B. Cosgrave and 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.

FORMULA

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

EXAMPLE

[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

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 *)

PROG

(SageMath)

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)]

(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

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).