A133221 -id:A133221 - cp's OEIS frontend

A055634 2-adic factorial function.

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

Michael Somos, Jun 06 2000

sign

Also known as Morita's 2-adic gamma function. - Harry Richman, Jul 26 2023

Serge Lang, Cyclotomic Fields I and II, Springer-Verlag, 1990, p. 315.

Kenny Lau, Table of n, a(n) for n = 0..806
Daniel Barsky, On Morita's p-adic Gamma function, Groupe de travail d'analyse ultramétrique, 5 (1977-1978), Talk no. 3, 6 p.
Wikipedia, P-adic gamma function.

Cf. A000142, A006882, A001147, A133221.

/* Based on Gauss factorial n_2!: */ k:=2; [IsZero(n) select 1 else (-1)^n*&*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // Bruno Berselli, Dec 10 2013

a[ n_] := If[ n < 0, 0, n! (-1)^n / (n - Mod[n, 2])!!]; (* Michael Somos, Jun 30 2018 *)
4[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - x) Exp[x^2/2], {x, 0, n}]]; (* Michael Somos, Jun 30 2018 *)

{a(n) = if( n<1, 1, -if( n%2, n * a(n-1), a(n-1)))};

a(n)=(-1)^n*(n=bitor(n-1,1))!/(n\2)!>>(n\2) \\ 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)
def A055634(n): return (-1)^n*Gauss_factorial(n, 2)
[A055634(n) for n in (0..28)]  # Peter Luschny, Oct 01 2012

a(2*n) = -a(2*n - 1) = (2*n - 1)!!

a(n) = (-1)^n*n!/A037223(n), A037223(n) = 2^floor(n/2)*floor(n/2)!. Exponential generating function: (1-x)*exp(x^2/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

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

Original entry on oeis.org

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.

A161736 Denominators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425, 46326420401234675625, 370882277949065911875, 5938020471163465810125

Offset: 2

Johannes W. Meijer, Jun 18 2009

The BG2 matrix coefficients, see also A008956, are defined by BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .
A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - Wolfdieter Lang, Nov 16 2016
Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - Matthew House, Dec 03 2024

			sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 2..830
R. Arratia, S. Garibaldi, and J. Kilian, Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process, arXiv:1310.7055 [math.PR], 2013.

Cf. Numerators A161737.
Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474, A278145.

[Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012
r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
seq(a(n), n = 1..21);  # Peter Luschny, Feb 12 2025

sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

{a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* Michael Somos, May 09 2011 */

a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - Michael Somos, May 09 2011
a(n) = A001902(2*n-3). - Mats Granvik, Nov 25 2018
a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 13 2025

A341847 Array read by antidiagonals: T(n,m) is the number of maximal matchings in the rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 1, 3, 10, 10, 3, 1, 1, 15, 40, 84, 40, 15, 1, 1, 15, 296, 852, 852, 296, 15, 1, 1, 105, 1576, 11580, 22368, 11580, 1576, 105, 1, 1, 105, 15352, 197640, 822528, 822528, 197640, 15352, 105, 1, 1, 945, 104000, 4314240, 38772864, 84961440, 38772864, 4314240, 104000, 945, 1

Offset: 0

Andrew Howroyd, Feb 21 2021

			Array begins:
=============================================================
n\m | 0  1    2      3        4           5             6
----+--------------------------------------------------------
  0 | 1  1    1      1        1           1             1 ...
  1 | 1  1    1      3        3          15            15 ...
  2 | 1  1    2     10       40         296          1576 ...
  3 | 1  3   10     84      852       11580        197640 ...
  4 | 1  3   40    852    22368      822528      38772864 ...
  5 | 1 15  296  11580   822528    84961440   12002446080 ...
  6 | 1 15 1576 197640 38772864 12002446080 5429866337280 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..275
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set
Eric Weisstein's World of Mathematics, Rook Graph

Rows n=1..4 are A133221(n+1), A281433, A341848, A341849.
Main diagonal is A289198.
Cf. A270227 (matchings), A297471, A341850 (maximum matchings).

T(n,m) = T(m,n).

A341850 Array read by antidiagonals: T(n,m) is the number of maximum matchings in the rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 1, 3, 4, 4, 3, 1, 1, 15, 16, 72, 16, 15, 1, 1, 15, 56, 132, 132, 56, 15, 1, 1, 105, 376, 7020, 2016, 7020, 376, 105, 1, 1, 105, 1912, 17280, 44928, 44928, 17280, 1912, 105, 1, 1, 945, 17984, 1920240, 1551744, 22615200, 1551744, 1920240, 17984, 945, 1

Offset: 0

Andrew Howroyd, Feb 21 2021

In the case that both m and n are odd a single vertex is not covered, otherwise the maximum matchings are perfect matchings.

			Array begins:
======================================================
n\m | 0  1   2     3       4         5           6
----+-------------------------------------------------
  0 | 1  1   1     1       1         1           1 ...
  1 | 1  1   1     3       3        15          15 ...
  2 | 1  1   2     4      16        56         376 ...
  3 | 1  3   4    72     132      7020       17280 ...
  4 | 1  3  16   132    2016     44928     1551744 ...
  5 | 1 15  56  7020   44928  22615200   243319680 ...
  6 | 1 15 376 17280 1551744 243319680 61903180800 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..527
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
Eric Weisstein's World of Mathematics, Rook Graph

Rows n=1..4 are A133221(n+1), A081919, A341851, A341852.
Main diagonal is A289197.
Cf. A270227 (matchings), A286070, A341847 (maximal matchings).

T(n,m) = T(m,n).

cp's OEIS Frontend

A133227 Incorrect duplicate of A133221.

Views

Author

Keywords

A055634 2-adic factorial function.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A161736 Denominators of the column sums of the BG2 matrix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A341847 Array read by antidiagonals: T(n,m) is the number of maximal matchings in the rook graph K_n X K_m.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A341850 Array read by antidiagonals: T(n,m) is the number of maximum matchings in the rook graph K_n X K_m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula