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A295864
a(n) = hypergeom([-n, -n], [1], 1) * n! / (floor(n/2)!)^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
0,2
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+[n/2])*hypergeom2F1(-n,1/2,1,1]*hypergeom2F1(-[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.
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}]
PROG
(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)]
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Feb 13 2018
STATUS
approved