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A380910 a(n) = denominator(n!! / (n - 1)!!).

%I A380910 #15 Feb 11 2025 04:41:29
%S A380910 1,1,1,2,3,8,5,16,35,128,63,256,231,1024,429,2048,6435,32768,12155,
%T A380910 65536,46189,262144,88179,524288,676039,4194304,1300075,8388608,
%U A380910 5014575,33554432,9694845,67108864,300540195,2147483648,583401555,4294967296,2268783825,17179869184
%N A380910 a(n) = denominator(n!! / (n - 1)!!).
%H A380910 Paolo Xausa, <a href="/A380910/b380910.txt">Table of n, a(n) for n = 0..1000</a>
%H A380910 <a href="https://dlmf.nist.gov/5.4#i">Gamma Function</a>, Digital library of mathematical functions, Feb. 2024.
%F A380910 r(n) = ((n/2)! / ((n - 1) / 2)!) * [sqrt(Pi) if n is even otherwise 2/sqrt(Pi)].
%F A380910 a(n) = denominator(r(n)).
%F A380910 a(n) = A004730(n-1). - _R. J. Mathar_, Feb 10 2025
%F A380910 a(n) = A006882(n-1)/A095987(n). - _R. J. Mathar_, Feb 10 2025
%p A380910 seq(denom(doublefactorial(n) / doublefactorial(n - 1)), n = 0..20);
%p A380910 # Alternative:
%p A380910 a := n -> denom((GAMMA(n/2 + 1) / GAMMA(n/2 + 1/2)) * ifelse(n::even, sqrt(Pi), 2/sqrt(Pi))):
%p A380910 seq(a(n), n = 0..35);
%t A380910 A380910[n_] := Denominator[n!!/(n - 1)!!]; Array[A380910, 50, 0] (* or *)
%t A380910 Denominator[FoldList[#2/# &, 1, Range[49]]] (* _Paolo Xausa_, Feb 11 2025 *)
%o A380910 (Python)
%o A380910 from fractions import Fraction
%o A380910 from functools import cache
%o A380910 @cache
%o A380910 def R(n: int) -> Fraction:
%o A380910     if n == 0: return Fraction(1, 1)
%o A380910     return Fraction(n, 1) / R(n - 1)
%o A380910 def aList(upto:int) -> list[int]:
%o A380910     return [R(n).denominator for n in range(upto + 1)]
%o A380910 print(aList(37))
%Y A380910 Cf. A380909 (numerator).
%Y A380910 Cf. A004730, A004731.
%K A380910 nonn,frac
%O A380910 0,4
%A A380910 _Peter Luschny_, Feb 09 2025