%I #21 Feb 14 2018 10:51:48

%S 1,2,12,120,420,7560,18480,480480,900900,30630600,46558512,1955457504,

%T 2498640144,124932007200,137680171200,7985449929600,7735904619300,

%U 510569704873800,441233078286000,32651247793164000,25467973278667920,2088373808850769440,1484298740174927040

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

%F a(2*n) = A000897(n).

%F a(n) = A000984(n) * A056040(n).

%F a(n) = (2*n)!/(n!*floor(n/2)!^2).

%F a(n) = (2^(2*n)*Gamma(n+1/2))/(sqrt(Pi)*Gamma(floor(n/2)+1)^2).

%F a(n) = multinomial([n/2], [n/2], n mod 2)*multinomial(n, n).

%F a(n) = 4^(n+[n/2])*hypergeom2F1(-n,1/2,1,1]*hypergeom2F1(-[n/2],(-1)^n/2,1,1].

%F 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 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 a := n -> binomial(2*n, n)*n!/iquo(n, 2)!^2: seq(a(n), n=0..22);

%t a[n_] := Multinomial[Quotient[n,2], Quotient[n,2], Mod[n,2]] Multinomial[n,n];

%t Table[a[n], {n, 0, 22}]

%o (Python)

%o def A295864():

%o r, c, n = 1, 1, 0

%o while True:

%o yield r * c

%o n += 1

%o c = c*(4*n-2)//n

%o r = (r*4)//n if n % 2 == 0 else r*n

%o a = A295864(); [next(a) for i in range(23)]

%Y Cf. A000897, A000984, A056040.

%K nonn

%O 0,2

%A _Peter Luschny_, Feb 13 2018