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A004730 Numerator of n!!/(n+1)!! (cf. A006882).

%I A004730 #51 Feb 09 2025 10:23:09
%S A004730 1,1,2,3,8,5,16,35,128,63,256,231,1024,429,2048,6435,32768,12155,
%T A004730 65536,46189,262144,88179,524288,676039,4194304,1300075,8388608,
%U A004730 5014575,33554432,9694845,67108864,300540195,2147483648,583401555,4294967296,2268783825
%N A004730 Numerator of n!!/(n+1)!! (cf. A006882).
%H A004730 T. D. Noe, <a href="/A004730/b004730.txt">Table of n, a(n) for n=0..300</a>
%H A004730 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 A004730 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.
%F A004730 Let y(m) = y(m-2) + 1/y(m-1) for m >= 2, with y(0)=y(1)=1. Then the denominator of y(n+1) equals the numerator of n!!/(n+1)!! for n >= 0, where the double factorials are given by A006882. [_Reinhard Zumkeller_, Dec 08 2011, as corrected in Cooper (2015)]
%t A004730 Numerator[#[[1]]/#[[2]]&/@Partition[Range[0,40]!!,2,1]] (* _Harvey P. Dale_, Jan 22 2013 *)
%t A004730 Numerator[CoefficientList[Series[(1 - Sqrt[1 - c^2] + ArcSin[c])/(c Sqrt[1 - c^2]),{c, 0, 39}], c]] (* _Eugene d'Eon_, Nov 01 2018 *)
%o A004730 (Haskell)
%o A004730 import Data.Ratio ((%), denominator)
%o A004730 a004730 n = a004730_list !! n
%o A004730 a004730_list = map denominator ggs where
%o A004730    ggs = 1 : 2 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
%o A004730 -- _Reinhard Zumkeller_, Dec 08 2011
%o A004730 (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ Numerator(DoubleFactorial(n) / DoubleFactorial(n+1)): n in [0..35]]; // _Vincenzo Librandi_, Dec 03 2018
%o A004730 (Python)
%o A004730 from sympy import gcd, factorial2
%o A004730 def A004730(n):
%o A004730     a, b = factorial2(n), factorial2(n+1)
%o A004730     return a//gcd(a,b) # _Chai Wah Wu_, Apr 03 2021
%o A004730 (PARI) f(n) = prod(i=0, (n-1)\2, n - 2*i); \\ A006882
%o A004730 a(n) = numerator(f(n)/f(n+1)); \\ _Michel Marcus_, Feb 09 2025
%Y A004730 Cf. A004731 (denominator), A006882 (double factorials).
%K A004730 nonn,frac
%O A004730 0,3
%A A004730 _N. J. A. Sloane_