Asymptotic analysis with factorial and exponential I'm solving a complexity question where I have:

$$ n!/2^n $$

The goal is to find an upper bound for this.

My idea is using the fact that: 
$$ n! = O(n^n)$$
$$ n!/2^n = O((n/2)^n) = O(n^n)$$

But is a correct upper bound, and if so is it the tightest upper bound that can be found? I know that n! is asymptotically larger than 2^n, but I'm struggling to do a tighter analysis.
Stirling's approximation states that
$$
n! \sim \sqrt{2\pi n} (n/e)^n.
$$
It follows that
$$
n!/2^n \sim \sqrt{2\pi n}(n/2e)^n = \Theta(\sqrt{n}(n/2e)^n).
$$