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A380909 a(n) = numerator(n!! / (n - 1)!!).

%I A380909 #27 Feb 11 2025 07:13:35
%S A380909 1,1,2,3,8,15,16,35,128,315,256,693,1024,3003,2048,6435,32768,109395,
%T A380909 65536,230945,262144,969969,524288,2028117,4194304,16900975,8388608,
%U A380909 35102025,33554432,145422675,67108864,300540195,2147483648,9917826435,4294967296,20419054425
%N A380909 a(n) = numerator(n!! / (n - 1)!!).
%C A380909 Perhaps surprisingly, the double factorial is also defined for n = -1. The reason for this is that (-1/2)! is well defined. See the first formula.
%H A380909 Paolo Xausa, <a href="/A380909/b380909.txt">Table of n, a(n) for n = 0..1000</a>
%H A380909 <a href="https://dlmf.nist.gov/5.4#i">Gamma Function</a>, Digital library of mathematical functions, Feb. 2024.
%F A380909 r(n) = ((n/2)! / ((n - 1) / 2)!) * [sqrt(Pi) if n is even otherwise 2/sqrt(Pi)].
%F A380909 r(n) = n / r(n - 1) for n >= 1.
%F A380909 r(n) ~ sqrt(n)*exp(1/(4*n))*(Pi/2)^(cos(Pi*n)/2).
%F A380909 Product_{k=0..n} r(k) = A006882(n).
%F A380909 a(n) = numerator(r(n)).
%F A380909 a(n) = A006882(n)/A095987(n). - _R. J. Mathar_, Feb 10 2025
%F A380909 a(n) = A004731(n+1). - _R. J. Mathar_, Feb 10 2025
%p A380909 seq(numer(doublefactorial(n) / doublefactorial(n - 1)), n = 0..20);
%p A380909 # Alternative:
%p A380909 a := n -> numer((GAMMA(n/2 + 1) / GAMMA(n/2 + 1/2)) * ifelse(n::even, sqrt(Pi), 2/sqrt(Pi))):
%p A380909 seq(a(n), n = 0..35);
%t A380909 A380909[n_] := Numerator[n!!/(n - 1)!!]; Array[A380909, 50, 0] (* or *)
%t A380909 Numerator[FoldList[#2/# &, 1, Range[49]]] (* _Paolo Xausa_, Feb 11 2025 *)
%o A380909 (Python)
%o A380909 from fractions import Fraction
%o A380909 from functools import cache
%o A380909 @cache
%o A380909 def R(n: int) -> Fraction:
%o A380909     if n == 0: return Fraction(1, 1)
%o A380909     return Fraction(n, 1) / R(n - 1)
%o A380909 def aList(upto:int) -> list[int]:
%o A380909     return [R(n).numerator for n in range(upto + 1)]
%o A380909 print(aList(35))
%Y A380909 Cf. A380910 (denominator), A006882, A095987, A004730, A004731.
%K A380909 nonn,frac
%O A380909 0,3
%A A380909 _Peter Luschny_, Feb 09 2025