Why does the square root of n! grow exponentially faster than exponential functions?
$\log(n!)=\Theta(n\log n)$, (see Stirling's approximation), hence 

$\frac{2^{O(n)}}{\sqrt{\left(\frac{n}{2}\right)!}}=2^{O(n)-\log\sqrt{\left(\frac{n}{2}\right)!}}=2^{O(n)-\Theta(n\log n)}=2^{-\left(\Theta(n\log n)-O(n)\right)}=2^{-\Omega(n)}$.