n | 1 | 2 | 3 | 4 | 5 n!! | 1 | 2 | 3 | 8 | 15

2^(n/2 + 1/4 (1 - cos(π n))) π^(1/4 (cos(π n) - 1)) Γ(n/2 + 1)

(no roots exist)

1 + 1/2 n (log(2) - gamma ) + 1/48 n^2 (6 gamma ^2 + 6 log^2(2) - 12 gamma log(2) + π^2 (1 + log(64) - 6 log(π))) + 1/96 n^3 (-2 gamma ^3 - gamma (6 log^2(2) + π^2 (1 + log(64) - 6 log(π))) + gamma ^2 log(64) + π^2 log(2) (1 + log(64) - 6 log(π)) + 2 (log^3(2) + polygamma(2, 1))) + (n^4 (20 gamma ^4 + 20 gamma ^2 (6 log^2(2) + π^2 (1 + log(64) - 6 log(π))) + 20 π^2 log^2(2) (1 + log(64) - 6 log(π)) + π^4 (3 + 60 log^2(π) + 20 log(2) (-3 + log(8) - 6 log(π)) + 60 log(π)) - 80 gamma ^3 log(2) + 20 (log^4(2) + log(16) polygamma(2, 1)) - 40 gamma (π^2 log(2) (1 + log(64) - 6 log(π)) + 2 (log^3(2) + polygamma(2, 1)))))/7680 + O(n^5) (Taylor series)

e^(1/2 n (log(n/2) - 1)) 2^(1/4 (2 n - cos(π n) + 1)) π^(1/4 cos(π n)) (π^(1/4) sqrt(n) + 1/6 π^(1/4) sqrt(1/n) + 1/72 π^(1/4) (1/n)^(3/2) - (139 π^(1/4) (1/n)^(5/2))/6480 - (571 π^(1/4) (1/n)^(7/2))/155520 + (163879 π^(1/4) (1/n)^(9/2))/6531840 + O((1/n)^(11/2)))

d/dn(n!!) = 1/4 n!! (2 polygamma(0, n/2 + 1) + π log(2/π) sin(π n) + log(4))

n!! = n/2 ! 2^(n/2) (π/2)^(1/4 (-1 + cos(n π)))

n!! = Γ(1 + n/2) 2^(n/2) (π/2)^(1/4 (-1 + cos(n π)))

n!! = (1)_(n/2) 2^(n/2) (π/2)^(1/4 (-1 + cos(n π)))

n!!∝2^(1/2 sin^2((n π)/2)) e^(-n/2) n^((1 + n)/2) π^(1/4 (1 + cos(n π))) (1 + sum_(k=1)^∞ sum_(j=1)^(2 k) ((-1/2)^j n^(-k) D_(2 (j + k), j))/((j + k)!)) for ((abs(arg(n))<π and abs(n)->∞ and D_(m, j) = (-1 + m) ((-2 + m) D_(-3 + m, -1 + j) + D_(-1 + m, j)) and D_(0, 0) = 1 and D_(m, 1) = (-1 + m)! and D_(m, j) = 0) for m<=3 j - 1)