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A237884 a(n) = (n!*m)/(m!*(m+1)!) where m = floor(n/2).

%I A237884 #25 Feb 25 2025 20:43:42
%S A237884 0,0,1,3,4,20,15,105,56,504,210,2310,792,10296,3003,45045,11440,
%T A237884 194480,43758,831402,167960,3527160,646646,14872858,2496144,62403600,
%U A237884 9657700,260757900,37442160,1085822640,145422675,4508102925,565722720,18668849760,2203961430
%N A237884 a(n) = (n!*m)/(m!*(m+1)!) where m = floor(n/2).
%F A237884 a(2*n) = A001791(n).
%F A237884 a(2*n+1) = A000917(n-1).
%F A237884 a(n) = n^(n mod 2)*binomial(2*floor(n/2), floor(n/2)-1).
%F A237884 a(n) = A162246(n, n+2) = n!/((n-ceiling((n+2)/2))!*floor((n+2)/2)!) if n > 1, otherwise 0.
%F A237884 a(n) = A056040(n)*floor(n/2)/(floor(n/2)+1).
%F A237884 a(n) + A056040(n) = A057977(n).
%F A237884 G.f.: -((p - 1 - x*(p - 1 + 2*x*(2*p - 3 + x*(3 + 4*x - 2*p))))/(2*x^2*p^3)), where p=sqrt(1-4*x^2). - _Benedict W. J. Irwin_, Aug 15 2016
%p A237884 A237884 := proc(n) m := iquo(n,2); (n!*m)/(m!*(m+1)!) end;
%p A237884 seq(A237884(n), n = 0..34);
%t A237884 CoefficientList[Series[-((-1 + Sqrt[1 - 4 x^2] -x (-1 + Sqrt[1 - 4 x^2] +
%t A237884 2 x (-3 + 2 Sqrt[1 - 4 x^2] +x (3 + 4 x - 2 Sqrt[1 - 4 x^2]))))/
%t A237884 (2 x^2 (1 - 4 x^2)^(3/2))), {x, 0, 30}], x] (* _Benedict W. J. Irwin_, Aug 15 2016 *)
%t A237884 Table[(n! #)/(#! (# + 1)!) &@ Floor[n/2], {n, 0, 34}] (* _Michael De Vlieger_, Aug 15 2016 *)
%o A237884 (Sage)
%o A237884 def A237884():
%o A237884     r, s, n = 1, 0, 0
%o A237884     while True:
%o A237884         yield s
%o A237884         n += 1
%o A237884         r *= 4/n if is_even(n) else n
%o A237884         s = r * (n//2)/(n//2+1)
%o A237884 a = A237884(); [next(a) for i in range(35)]
%K A237884 nonn
%O A237884 0,4
%A A237884 _Peter Luschny_, Feb 14 2014