%I #48 Sep 23 2022 16:17:13
%S 1,1,1,2,3,8,15,16,35,128,315,256,693,1024,3003,2048,6435,32768,
%T 109395,65536,230945,262144,969969,524288,2028117,4194304,16900975,
%U 8388608,35102025,33554432,145422675,67108864
%N Denominator of n!!/(n+1)!! (cf. A006882).
%C Also numerator of rational part of Haar measure on Grassmannian space G(n,1).
%C Also rational part of numerator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A036039).
%C Let x(m) = x(m-2) + 1/x(m-1) for m >= 3, with x(1)=x(2)=1. Then the numerator of
%C 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).
%D D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.
%H T. D. Noe, <a href="/A004731/b004731.txt">Table of n, a(n) for n=0..302</a>
%H Joseph E. Cooper III, <a href="http://arxiv.org/abs/1510.00399">A recurrence for an expression involving double factorials</a>, arXiv:1510.00399 [math.CO], 2015.
%H Svante Janson, <a href="http://www2.math.uu.se/~svante/papers/sj114.pdf">On the traveling fly problem</a>, Graph Theory Notes of New York Vol. XXXI, 17, 1996.
%e 1, 1, (1/2)*Pi, 2, (3/4)*Pi, 8/3, (15/16)*Pi, 16/5, (35/32)*Pi, 128/35, (315/256)*Pi, ...
%e 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), ...
%p 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;
%p f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));
%t Table[ Denominator[ (n-2)!! / (n-1)!! ], {n, 0, 31}] (* _Jean-François Alcover_, Jul 16 2012 *)
%t Denominator[#[[1]]/#[[2]]&/@Partition[Range[-2,40]!!,2,1]] (* _Harvey P. Dale_, Nov 27 2014 *)
%o (Haskell)
%o import Data.Ratio ((%), numerator)
%o a004731 0 = 1
%o a004731 n = a004731_list !! n
%o a004731_list = map numerator ggs where
%o ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
%o -- _Reinhard Zumkeller_, Dec 08 2011
%o (Python)
%o from sympy import gcd, factorial2
%o def A004731(n):
%o if n <= 1:
%o return 1
%o a, b = factorial2(n-2), factorial2(n-1)
%o return b//gcd(a,b) # _Chai Wah Wu_, Apr 03 2021
%Y Cf. A001803, A004730, A006882 (double factorials), A036069.
%Y Cf. A036039, A046161, A001790, A001803, A101926.
%K nonn,easy,nice,frac
%O 0,4
%A _N. J. A. Sloane_