OFFSET
0,4
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
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.
FORMULA
a(n) = Product_{k=1..n} A000265(k).
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
EXAMPLE
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)
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 *)
PROG
(Magma) [ Denominator(2^n/Factorial(n)): n in [0..25] ]; // Klaus Brockhaus, Mar 10 2011
(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
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
N. J. A. Sloane, Feb 05 2000
EXTENSIONS
New name (from Amarnath Murthy) by Charles R Greathouse IV, Jul 23 2015
STATUS
approved