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A049606 Largest odd divisor of n!.

%I A049606 #87 May 12 2023 09:09:28
%S A049606 1,1,1,3,3,15,45,315,315,2835,14175,155925,467775,6081075,42567525,
%T A049606 638512875,638512875,10854718875,97692469875,1856156927625,
%U A049606 9280784638125,194896477400625,2143861251406875,49308808782358125,147926426347074375,3698160658676859375
%N A049606 Largest odd divisor of n!.
%C A049606 Original name: Denominator of 2^n/n!.
%C A049606 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
%C A049606 a(n) is also the number of binomial heaps with n nodes. - _Zhujun Zhang_, Jun 16 2019
%C A049606 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
%H A049606 T. D. Noe, <a href="/A049606/b049606.txt">Table of n, a(n) for n = 0..100</a>
%H A049606 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/1376586">On the number of Sylow subgroups in Symmetric Group</a>
%H A049606 Zhujun Zhang, <a href="https://www.researchgate.net/publication/333261503">A Note on Counting Binomial Heaps</a>, ResearchGate, June 2019.
%F A049606 a(n) = Product_{k=1..n} A000265(k).
%F A049606 a(n) = A000265(A000142(n)). - _Reinhard Zumkeller_, Apr 09 2004
%F A049606 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
%F A049606 a(n) = denominator([t^n] 1/(tanh(t)-1)). - _Peter Luschny_, Aug 04 2011
%F A049606 a(n) = n!/2^A011371(n). - _Robert Israel_, Jul 23 2015
%F A049606 From _Zhujun Zhang_, Jun 16 2019: (Start)
%F A049606 a(n) = n!/A060818(n).
%F A049606 E.g.f.: Product_{k>=0} (1 + x^(2^k) / 2^(2^k - 1)).
%F A049606 (End)
%F A049606 log a(n) = n log n - (1 + log 2)n + Θ(log n). - _Charles R Greathouse IV_, Feb 12 2022
%e A049606 From _John M. Campbell_, May 28 2011: (Start)
%e A049606 The numerator of the permanent of the following 5 X 5 matrix is equal to a(5):
%e A049606 |  1/4  -1/4  -1/2  -1/4   1/4 |
%e A049606 | -1/4   1/4   1/2   1/4  -1/4 |
%e A049606 | -1/2   1/2    1    1/2  -1/2 |
%e A049606 | -1/4   1/4   1/2   1/4  -1/4 |
%e A049606 |  1/4  -1/4  -1/2  -1/4   1/4 | (End)
%p A049606 f:= n-> n! * 2^(add(i,i=convert(n,base,2))-n); # _Peter Luschny_, May 02 2009
%p A049606 seq (denom (coeff (series(1/(tanh(t)-1), t, 30), t, n)), n=0..25); # _Peter Luschny_, Aug 04 2011
%p A049606 seq(numer(n!/2^n), n=0..100); # _Robert Israel_, Jul 23 2015
%t A049606 Denominator[Table[(2^n)/n!,{n,0,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 03 2011*)
%t A049606 Table[Last[Select[Divisors[n!],OddQ]],{n,0,30}] (* _Harvey P. Dale_, Jul 24 2016 *)
%t A049606 Table[n!/2^IntegerExponent[n!,2], {n,1,30}] (* _Clark Kimberling_, Oct 22 2016 *)
%o A049606 (Magma) [ Denominator(2^n/Factorial(n)): n in [0..25] ]; // _Klaus Brockhaus_, Mar 10 2011
%o A049606 (PARI) A049606(n)=local(f=n!);f/2^valuation(f,2); \\ _Joerg Arndt_, Apr 22 2011
%o A049606 (Python 3.10+)
%o A049606 from math import factorial
%o A049606 def A049606(n): return factorial(n)>>n-n.bit_count() # _Chai Wah Wu_, Jul 11 2022
%Y A049606 Numerators of 2^n/n! give A001316. Cf. A000680, A008977, A139541.
%Y A049606 Factor of A160481. - _Johannes W. Meijer_, May 24 2009
%Y A049606 Equals A003148 divided by A123746. - _Johannes W. Meijer_, Nov 23 2009
%Y A049606 Different from A160624.
%Y A049606 Cf. A011371.
%K A049606 nonn,frac,easy
%O A049606 0,4
%A A049606 _N. J. A. Sloane_, Feb 05 2000
%E A049606 New name (from _Amarnath Murthy_) by _Charles R Greathouse IV_, Jul 23 2015