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A049606
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Largest odd divisor of n!.
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30
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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
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OFFSET
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0,4
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COMMENTS
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Original name: Denominator of 2^n/n!.
a(n) = A000265(A000142(n)). - Reinhard Zumkeller, Apr 09 2004
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
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
Zhujun Zhang, A Note on Counting Binomial Heaps, ResearchGate, June 2019.
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FORMULA
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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)
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Numerators 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.
Sequence in context: A067655 A209430 A160624 * A046126 A143257 A089403
Adjacent sequences: A049603 A049604 A049605 * A049607 A049608 A049609
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KEYWORD
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nonn,frac,easy
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AUTHOR
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N. J. A. Sloane, Feb 05 2000
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EXTENSIONS
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New name (from Amarnath Murthy) by Charles R Greathouse IV, Jul 23 2015
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STATUS
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approved
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