A232980 -id:A232980 - cp's OEIS frontend

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

Peter Luschny, Oct 01 2012

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.

			[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

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.

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.

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

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 *)

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

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)]

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

A232985 The Gauss factorial n_11!.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 3628800, 43545600, 566092800, 7925299200, 118879488000, 1902071808000, 32335220736000, 582033973248000, 11058645491712000, 221172909834240000, 4644631106519040000, 4644631106519040000, 106826515449937920000, 2563836370798510080000

Offset: 0

N. J. A. Sloane, Dec 08 2013

nonn

The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.

J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.
J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

The Gauss factorials n_1!, n_2!, n_3!, n_5!, n_6!, n_7!, n_10!, n_11! are A000142, A055634, A232980-A232985 respectively.

k:=11; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // Bruno Berselli, Dec 10 2013

Gf:=proc(N,n) local j,k; k:=1;
for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
f:=n->[seq(Gf(N,n),N=0..40)];
f(11);

A232981 The Gauss factorial n_5!.

Original entry on oeis.org

1, 1, 2, 6, 24, 24, 144, 1008, 8064, 72576, 72576, 798336, 9580032, 124540416, 1743565824, 1743565824, 27897053184, 474249904128, 8536498274304, 162193467211776, 162193467211776, 3406062811447296, 74933381851840512, 1723467782592331776, 41363226782215962624, 41363226782215962624

Offset: 0

N. J. A. Sloane, Dec 08 2013

nonn

The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.

Robert Israel, Table of n, a(n) for n = 0..542
J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.
J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

The Gauss factorials n_1!, n_2!, n_3!, n_5!, n_6!, n_7!, n_10!, n_11! are A000142, A055634, A232980-A232985 respectively.

k:=5; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // Bruno Berselli, Dec 10 2013

Gf:=proc(N,n) local j,k; k:=1;
for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
f:=n->[seq(Gf(N,n),N=0..40)];
f(5);

Table[n!/(5^#*#!) &@ Floor[n/5], {n, 0, 25}] (* Michael De Vlieger, Mar 06 2017 *)

From Robert Israel, Mar 06 2017: (Start)
a(n) = a(n-1) if 5 | n; otherwise n*a(n-1).
a(n) = n!/(5^floor(n/5)*floor(n/5)!). (End)

A232982 The Gauss factorial n_6!.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 5, 35, 35, 35, 35, 385, 385, 5005, 5005, 5005, 5005, 85085, 85085, 1616615, 1616615, 1616615, 1616615, 37182145, 37182145, 929553625, 929553625, 929553625, 929553625, 26957055125, 26957055125, 835668708875, 835668708875, 835668708875, 835668708875, 29248404810625, 29248404810625

Offset: 0

N. J. A. Sloane, Dec 08 2013

nonn

The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.

J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.
J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

The Gauss factorials n_1!, n_2!, n_3!, n_5!, n_6!, n_7!, n_10!, n_11! are A000142, A055634, A232980-A232985 respectively.

k:=6; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..40]]; // Bruno Berselli, Dec 10 2013

Gf:=proc(N,n) local j,k; k:=1;
for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
f:=n->[seq(Gf(N,n),N=0..40)];
f(6);

A232983 The Gauss factorial n_7!.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 720, 5760, 51840, 518400, 5702400, 68428800, 889574400, 889574400, 13343616000, 213497856000, 3629463552000, 65330343936000, 1241276534784000, 24825530695680000, 24825530695680000, 546161675304960000, 12561718532014080000, 301481244768337920000, 7537031119208448000000

Offset: 0

N. J. A. Sloane, Dec 08 2013

nonn

The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.

J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.
J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

The Gauss factorials n_1!, n_2!, n_3!, n_5!, n_6!, n_7!, n_10!, n_11! are A000142, A055634, A232980-A232985 respectively.

k:=7; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // Bruno Berselli, Dec 10 2013

Gf:=proc(N,n) local j,k; k:=1;
for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
f:=n->[seq(Gf(N,n),N=0..40)];
f(7);

A232984 The Gauss factorial n_10!.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 3, 21, 21, 189, 189, 2079, 2079, 27027, 27027, 27027, 27027, 459459, 459459, 8729721, 8729721, 183324141, 183324141, 4216455243, 4216455243, 4216455243, 4216455243, 113844291561, 113844291561, 3301484455269, 3301484455269, 102346018113339, 102346018113339, 3377418597740187

Offset: 0

N. J. A. Sloane, Dec 08 2013

nonn

The Gauss factorial n_k! is defined to be Product_{1<=j<=n, gcd(j,k)=1} j.

J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.
J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

The Gauss factorials n_1!, n_2!, n_3!, n_5!, n_6!, n_7!, n_10!, n_11! are A000142, A055634, A232980-A232985 respectively.

k:=10; [IsZero(n) select 1 else &*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..40]]; // Bruno Berselli, Dec 10 2013

Gf:=proc(N,n) local j,k; k:=1;
for j from 1 to N do if gcd(j,n)=1 then k:=j*k; fi; od; k; end;
f:=n->[seq(Gf(N,n),N=0..40)];
f(10);

cp's OEIS Frontend

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

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A232985 The Gauss factorial n_11!.

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A232981 The Gauss factorial n_5!.

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A232982 The Gauss factorial n_6!.

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A232983 The Gauss factorial n_7!.

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A232984 The Gauss factorial n_10!.

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