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A295864 a(n) = hypergeom([-n, -n], [1], 1) * n! / (floor(n/2)!)^2.

%I A295864 #24 Mar 15 2025 02:36:41
%S A295864 1,2,12,120,420,7560,18480,480480,900900,30630600,46558512,1955457504,
%T A295864 2498640144,124932007200,137680171200,7985449929600,7735904619300,
%U A295864 510569704873800,441233078286000,32651247793164000,25467973278667920,2088373808850769440,1484298740174927040
%N A295864 a(n) = hypergeom([-n, -n], [1], 1) * n! / (floor(n/2)!)^2.
%F A295864 a(2*n) = A000897(n).
%F A295864 a(n) = A000984(n) * A056040(n).
%F A295864 a(n) = (2*n)!/(n!*floor(n/2)!^2).
%F A295864 a(n) = (2^(2*n)*Gamma(n+1/2))/(sqrt(Pi)*Gamma(floor(n/2)+1)^2).
%F A295864 a(n) = multinomial([n/2], [n/2], n mod 2)*multinomial(n, n).
%F A295864 a(n) = 4^(n+floor(n/2))*hypergeom([-n,1/2],[1],1)*hypergeom([-floor(n/2),(-1)^n/2],[1],1).
%F A295864 a(n) = c(n)*8^n*Pochhammer(1/4, [n/2])*Pochhammer(3/4, [n/2])/[n/2]!^2 where c(n) = 1 if n is even else c(n) = (2*n-1)/4.
%F A295864 a(n) ~ (8^n/(sqrt(2)*Pi*n))*c(n) where c(n) = 2 - 3/(4*n) if n is even else c(n) = n + 1/8.
%p A295864 a := n -> binomial(2*n, n)*n!/iquo(n, 2)!^2: seq(a(n), n=0..22);
%t A295864 a[n_] := Multinomial[Quotient[n,2], Quotient[n,2], Mod[n,2]] Multinomial[n,n];
%t A295864 Table[a[n], {n, 0, 22}]
%o A295864 (Python)
%o A295864 def A295864():
%o A295864     r, c, n = 1, 1, 0
%o A295864     while True:
%o A295864         yield r * c
%o A295864         n += 1
%o A295864         c = c*(4*n-2)//n
%o A295864         r = (r*4)//n if n % 2 == 0 else r*n
%o A295864 a = A295864(); [next(a) for i in range(23)]
%Y A295864 Cf. A000897, A000984, A056040.
%K A295864 nonn
%O A295864 0,2
%A A295864 _Peter Luschny_, Feb 13 2018