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A049606 Largest odd divisor of n!. 30
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

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)

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

CROSSREFS

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

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

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Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)