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

1,4

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, 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.

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

(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)]
(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).

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.

Sequence in context: A216914 A216917 A320637 * A152656 A096162 A333144

Adjacent sequences: A216916 A216917 A216918 * A216920 A216921 A216922

Keyword

nonn,tabl

Author

Peter Luschny, Oct 01 2012

Status

approved