A004731 Denominator of n!!/(n+1)!! (cf. A006882).

1, 1, 1, 2, 3, 8, 15, 16, 35, 128, 315, 256, 693, 1024, 3003, 2048, 6435, 32768, 109395, 65536, 230945, 262144, 969969, 524288, 2028117, 4194304, 16900975, 8388608, 35102025, 33554432, 145422675, 67108864

Offset

0,4

Comments

Also numerator of rational part of Haar measure on Grassmannian space G(n,1).

Also rational part of numerator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A036039).

Let x(m) = x(m-2) + 1/x(m-1) for m >= 3, with x(1)=x(2)=1. Then the numerator of

x(n+2) equals the denominator of n!!/(n+1)!! for n >= 0, where the double factorials are given by A006882. - Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 07 2010, as corrected in Cooper (2015).

References

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

Links

T. D. Noe, Table of n, a(n) for n=0..302

Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.

Svante Janson, On the traveling fly problem, Graph Theory Notes of New York Vol. XXXI, 17, 1996.

Example

1, 1, (1/2)*Pi, 2, (3/4)*Pi, 8/3, (15/16)*Pi, 16/5, (35/32)*Pi, 128/35, (315/256)*Pi, ...
The sequence Gamma(n/2+1)/Gamma(n/2+1/2), n >= 0, begins 1/Pi^(1/2), 1/2*Pi^(1/2), 2/Pi^(1/2), 3/4*Pi^(1/2), 8/3/Pi^(1/2), 15/16*Pi^(1/2), 16/5/Pi^(1/2), ...

Maple

if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1, k)/4^k else k := (n-1)/2; 4^k/binomial(2*k, k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));

Mathematica

Table[ Denominator[ (n-2)!! / (n-1)!! ], {n, 0, 31}] (* Jean-François Alcover, Jul 16 2012 *)
Denominator[#[[1]]/#[[2]]&/@Partition[Range[-2, 40]!!, 2, 1]] (* Harvey P. Dale, Nov 27 2014 *)

Prog

(Haskell)
import Data.Ratio ((%), numerator)
a004731 0 = 1
a004731 n = a004731_list !! n
a004731_list = map numerator ggs where
   ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
-- Reinhard Zumkeller, Dec 08 2011
(Python)
from sympy import gcd, factorial2
def A004731(n):
    if n <= 1:
        return 1
    a, b = factorial2(n-2), factorial2(n-1)
    return b//gcd(a, b) # Chai Wah Wu, Apr 03 2021

Crossrefs

Cf. A001803, A004730, A006882 (double factorials), A036069.

Cf. A036039, A046161, A001790, A001803, A101926.

Sequence in context: A069752 A265694 A238778 * A135354 A240974 A356879

Adjacent sequences: A004728 A004729 A004730 * A004732 A004733 A004734

Keyword

nonn,easy,nice,frac

Author

N. J. A. Sloane

Status

approved