A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

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, 300540195, 2147483648, 9917826435, 4294967296, 20419054425

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• frac

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).

Numerator of (n-1)/( (n-2)/( .../1)), with an empty fraction taken to be 1. - Flávio V. Fernandes, Jan 31 2025

Examples

			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), ...
		

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.

Crossrefs

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

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

Programs

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

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));
#
[1, seq(denom(doublefactorial(n-2)/doublefactorial(n-1)), n = 1..36)]; # Peter Luschny, Feb 09 2025

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 *)
Join[{1},Table[Numerator[(n/2-1/2)!/((n/2-1)!Sqrt[Pi])], {n,1,31}]] (* Peter Luschny, Feb 08 2025 *)

PARI

f(n) = prod(i=0, (n-1)\2, n - 2*i); \\ A006882
a(n) = if (n==0, 1, denominator(f(n-2)/f(n-1))); \\ Michel Marcus, Feb 08 2025

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

Extensions

Name corrected by Michel Marcus, Feb 08 2025