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

1, 2, 12, 120, 420, 7560, 18480, 480480, 900900, 30630600, 46558512, 1955457504, 2498640144, 124932007200, 137680171200, 7985449929600, 7735904619300, 510569704873800, 441233078286000, 32651247793164000, 25467973278667920, 2088373808850769440, 1484298740174927040

Offset: 0

Author

Peter Luschny, Feb 13 2018

Keywords

• nonn

Crossrefs

Cf. A000897, A000984, A056040.

Programs

Maple

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

Mathematica

a[n_] := Multinomial[Quotient[n,2], Quotient[n,2], Mod[n,2]] Multinomial[n,n];
Table[a[n], {n, 0, 22}]

Python

def A295864():
    r, c, n = 1, 1, 0
    while True:
        yield r * c
        n += 1
        c = c*(4*n-2)//n
        r = (r*4)//n if n % 2 == 0 else r*n
a = A295864(); [next(a) for i in range(23)]

Formula

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

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

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

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

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

a(n) = 4^(n+floor(n/2))*hypergeom([-n,1/2],[1],1)*hypergeom([-floor(n/2),(-1)^n/2],[1],1).

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.

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.