A055634 -id:A055634 - 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.

A133221 A001147 with each term repeated.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945, 10395, 10395, 135135, 135135, 2027025, 2027025, 34459425, 34459425, 654729075, 654729075, 13749310575, 13749310575, 316234143225, 316234143225, 7905853580625, 7905853580625, 213458046676875, 213458046676875

Offset: 0

N. J. A. Sloane, Oct 13 2007

nonn

Normally such sequences are excluded from the OEIS, but I have made an exception for this one because so many variants of it have occurred in recent submissions.
For n>=2, a(n) = product of odd positive integers <=(n-1). - Jaroslav Krizek, Mar 21 2009
a(n) is, for n>=3, the number of way to choose floor((n-1)/2) disjoint pairs of items from n-1 items. It is then a fortiori the size of the conjugacy class of the reversal permutation [n-1,n-2,n-3,...,3,2,1]=(1 n-1)(2 n-2)(3 n-3)... in the symmetric group on n-1 elements. - Karl-Dieter Crisman, Nov 03 2009

G. C. Greubel, Table of n, a(n) for n = 0..800

Cf. A006882, A055634.

Appears in A161736. - Johannes W. Meijer, Jun 18 2009

f[x_] := E^(x^2/2) + Sqrt[Pi/2]*Erfi[x/Sqrt[2]]; CoefficientList[ Series[f[x], {x, 0, 29}], x]*Range[0, 29]! (* Jean-François Alcover, Sep 25 2012, after Sergei N. Gladkovskii *)
Table[(n - 1 - Mod[n, 2])!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)
Table[((2 n + (-1)^n - 3)/2)!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)

a(n) = my(k = (2*n + (-1)^n - 3)/2); prod(i=0, (k-1)\2, k - 2*i) \\ Iain Fox, Dec 31 2017

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A133221(n): return Gauss_factorial(n-1, 2)
[A133221(n) for n in (0..29)]  # Peter Luschny, Oct 01 2012

E.g.f.: x*U(0)  where U(k)= 1 + (2*k+1)/(x - x^4/(x^3 + (2*k+2)*(2*k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 25 2012
G.f.: 1+x*G(0), where G(k)= 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
a(n) = (2*floor(n/2)-1)!! = (n-1-(n mod 2))!!. - Alois P. Heinz, Sep 24 2024

A232980 The Gauss factorial n_3!.

Original entry on oeis.org

1, 1, 2, 2, 8, 40, 40, 280, 2240, 2240, 22400, 246400, 246400, 3203200, 44844800, 44844800, 717516800, 12197785600, 12197785600, 231757926400, 4635158528000, 4635158528000, 101973487616000, 2345390215168000, 2345390215168000, 58634755379200000, 1524503639859200000, 1524503639859200000

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:=3; [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(3);

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);

A076051 Sum of product of odd numbers <= n and the product of even numbers <= n.

Original entry on oeis.org

2, 3, 5, 11, 23, 63, 153, 489, 1329, 4785, 14235, 56475, 181215, 780255, 2672145, 12348945, 44781345, 220253985, 840523635, 4370620275, 17465201775, 95498916975, 397983749625, 2278224696825, 9867844134225, 58917607974225

Offset: 1

Emrehan Halici (emrehan(AT)halici.com.tr), Oct 30 2002

G. C. Greubel, Table of n, a(n) for n = 1..800

Cf. A037223, A055634, A060696.

A037223[n_] := 2^(Floor[n/2])*(Floor[n/2])!; Table[A037223[n] + n!/A037223[n] , {n,1,50}] (* G. C. Greubel, May 23 2017 *)
With[{nn = 25}, CoefficientList[Series[1 + x + (1 + x + x^2) *(Exp[x^2/2] *(1 + Sqrt[Pi/2]*Erf[x/Sqrt[2]])), {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 25 2017 *)

for(n=1, 50, print1(2^(floor(n/2))*(floor(n/2))! + n!/(2^(floor(n/2))*(floor(n/2))!), ", ")) \\ G. C. Greubel, May 23 2017

a(n) = o(n)+ e(n) where; o(n)=the product of odd numbers from 1 to n e(n)=the product of even numbers from 2 to n.
From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002: (Start)
a(n) = A060696(n+1).
a(n) = A037223(n) + abs(A055634(n)).
a(n) = A037223(n) + n! / A037223(n), where A037223(n) = 2^floor(n/2) * floor(n/2)!, for n>=2.
a(1)=2, a(2)=3, a(3)=5, a(n) = (n-1)*a(n-2) + (n-2)!! for n >= 4.
E.g.f.: 1 + x + (1+x+x^2)*(exp(x^2/2)*(1+sqrt(Pi/2)*erf(x/sqrt(2)))), where erf denotes the error function. (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

a(1) corrected by G. C. Greubel, May 23 2017

A185021 a(n) = h(1)h(2)...h(n), where h(i) = i/[g(i/2)g(i/4)g(i/8)...] and g(x) = x if x is an integer and g(x) = 1 otherwise.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 120, 840, 840, 7560, 15120, 166320, 110880, 1441440, 2882880, 43243200, 10810800, 183783600, 367567200, 6983776800, 2793510720, 58663725120, 117327450240, 2698531355520, 299836817280, 7495920432000, 14991840864000, 404779703328000, 115651343808000, 3353888970432000, 6707777940864000

Offset: 0

Bakir FARHI, Jan 22 2012

nonn

Although h(i) is not necessarily an integer, a(n) is.

Alois P. Heinz, Table of n, a(n) for n = 0..1662
B. Farhi, A study of a curious arithmetic function, arXiv:1004.2269 [math.NT], April 13 2010.
Bakir Farhi, A Study of a Curious Arithmetic Function, Journal of Integer Sequences, Vol. 15 (2012), #12.3.1.

Cf. A185275, A055634.

a:= proc(n) option remember; `if`(n<1, 1, h(n)*a(n-1)) end:
h:= i-> i/mul((t->`if`(t::integer, t, 1))((i/2^j)), j=1..ilog2(i)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 18 2018

a[n_] := a[n] = If[n<1, 1, h[n] a[n-1]];
h[i_] := i/Product[If[IntegerQ[#], #, 1]&[i/2^j], {j, 1, Log[2, i]}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 13 2018, after Alois P. Heinz *)

Edited by N. J. A. Sloane, Apr 10 2012

a(0)=1 prepended by Alois P. Heinz, Oct 18 2018

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)

A185275 Products of the first terms of the arithmetic sequence f(n) defined by f(2^k l) = l^{1 - k} (for k a nonnegative integer and l a positive odd integer).

Original entry on oeis.org

1, 1, 3, 3, 15, 15, 105, 105, 945, 945, 10395, 3465, 45045, 45045, 675675, 675675, 11486475, 11486475, 218243025, 43648605, 916620705, 916620705, 21082276215, 2342475135, 58561878375, 58561878375, 1581170716125, 225881530875, 6550564395375, 6550564395375

Offset: 0

Bakir FARHI, Jan 21 2012

nonn

Note that f(n) is not always an integer (for example f(12) = 1/3) but Farhi showed in his paper that the product Product_{i = 1..n} f(i) is always an integer.

B. Farhi, A study of a curious arithmetic function, arXiv:1004.2269 [math.NT], April 13 2010.
Bakir Farhi, A Study of a Curious Arithmetic Function, Journal of Integer Sequences, Vol. 15 (2012), #12.3.1.

Cf. A185021, A055634.

G.f.: G(0)/x -1/x, where G(k)= 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013

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);

cp's OEIS Frontend

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

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A133221 A001147 with each term repeated.

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A232980 The Gauss factorial n_3!.

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A232985 The Gauss factorial n_11!.

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A076051 Sum of product of odd numbers <= n and the product of even numbers <= n.

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A185021 a(n) = h(1)*h(2)*...*h(n), where h(i) = i/[g(i/2)*g(i/4)*g(i/8)*...] and g(x) = x if x is an integer and g(x) = 1 otherwise.

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A232981 The Gauss factorial n_5!.

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A185021 a(n) = h(1)h(2)...h(n), where h(i) = i/[g(i/2)g(i/4)g(i/8)...] and g(x) = x if x is an integer and g(x) = 1 otherwise.