A049606 Largest odd divisor of n!.

1, 1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 638512875, 638512875, 10854718875, 97692469875, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 49308808782358125, 147926426347074375, 3698160658676859375

Offset: 0

Author

N. J. A. Sloane, Feb 05 2000

Keywords

• nonn
• frac
• easy

Comments

Original name: Denominator of 2^n/n!.

For positive n, a(n) equals the numerator of the permanent of the n X n matrix whose (i,j)-entry is cos(i*Pi/3)*cos(j*Pi/3) (see example below). - John M. Campbell, May 28 2011

a(n) is also the number of binomial heaps with n nodes. - Zhujun Zhang, Jun 16 2019

a(n) is the number of 2-Sylow subgroups of the symmetric group S_n (see the Mathematics Stack Exchange link below). - Jianing Song, Nov 11 2022

Examples

			From _John M. Campbell_, May 28 2011: (Start)
The numerator of the permanent of the following 5 X 5 matrix is equal to a(5):
|  1/4  -1/4  -1/2  -1/4   1/4 |
| -1/4   1/4   1/2   1/4  -1/4 |
| -1/2   1/2    1    1/2  -1/2 |
| -1/4   1/4   1/2   1/4  -1/4 |
|  1/4  -1/4  -1/2  -1/4   1/4 | (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Mathematics Stack Exchange, On the number of Sylow subgroups in Symmetric Group
• Zhujun Zhang, A Note on Counting Binomial Heaps, ResearchGate, June 2019.

Crossrefs

Numerators of 2^n/n! give A001316. Cf. A000680, A008977, A139541.

Factor of A160481. - Johannes W. Meijer, May 24 2009

Equals A003148 divided by A123746. - Johannes W. Meijer, Nov 23 2009

Different from A160624.

Cf. A011371.

Programs

Magma

[ Denominator(2^n/Factorial(n)): n in [0..25] ]; // Klaus Brockhaus, Mar 10 2011

Maple

f:= n-> n! * 2^(add(i,i=convert(n,base,2))-n); # Peter Luschny, May 02 2009
seq (denom (coeff (series(1/(tanh(t)-1), t, 30), t, n)), n=0..25); # Peter Luschny, Aug 04 2011
seq(numer(n!/2^n), n=0..100); # Robert Israel, Jul 23 2015

Mathematica

Denominator[Table[(2^n)/n!,{n,0,40}]] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011*)
Table[Last[Select[Divisors[n!],OddQ]],{n,0,30}] (* Harvey P. Dale, Jul 24 2016 *)
Table[n!/2^IntegerExponent[n!,2], {n,1,30}] (* Clark Kimberling, Oct 22 2016 *)

PARI

A049606(n)=local(f=n!);f/2^valuation(f,2); \\ Joerg Arndt, Apr 22 2011
(Python 3.10+)
from math import factorial
def A049606(n): return factorial(n)>>n-n.bit_count() # Chai Wah Wu, Jul 11 2022

Formula

a(n) = Product_{k=1..n} A000265(k).

a(n) = A000265(A000142(n)). - Reinhard Zumkeller, Apr 09 2004

a(n) = numerator(2*Sum_{i>=1} (-1)^i*(1-zeta(n+i+1)) * (Product_{j=1..n} i+j)). - Gerry Martens, Mar 10 2011

a(n) = denominator([t^n] 1/(tanh(t)-1)). - Peter Luschny, Aug 04 2011

a(n) = n!/2^A011371(n). - Robert Israel, Jul 23 2015

From Zhujun Zhang, Jun 16 2019: (Start)

a(n) = n!/A060818(n).

E.g.f.: Product_{k>=0} (1 + x^(2^k) / 2^(2^k - 1)).

(End)

log a(n) = n log n - (1 + log 2)n + Θ(log n). - Charles R Greathouse IV, Feb 12 2022

Extensions

New name (from Amarnath Murthy) by Charles R Greathouse IV, Jul 23 2015