\frac{\sqrt{\pi} \Gamma(n+1/2)}{\Gamma(n+1)} = \frac{\sqrt{\pi} (n-1/2)(n-3/2)...3/2 \Gamma(1/2)}{n!} = \frac{\pi(2n-1)!!}{2^n n!} = \frac{\pi(2n-1)!!}{(2n)!!}