Asymptotic relation between specific binomial coefficient and exponential function I need to determine the asymptotic relationship between the functions:
$$f_1(n)={n\choose{\lfloor{n\over{2}}\rfloor}}, f_2(n)=7^{\sqrt{n}}$$
(I'm going to just assume $n$ is always even.)
I've convinced myself that $f_2(n) = o\left(f_1(n)\right)$ using the following argument:
$${n\choose{n\over 2}}={{n-2}\choose{{n\over 2}-1}}⋅{{n(n-1)}\over{\left({n\over 2}\right)^2}}=f_1(n-2)⋅4\left(1-{1\over n}\right)$$
The expression $\left(1 - {1 \over n}\right)$ tends to $1$ as $n→∞$, so $f_1(n)$ eventually grows about as fast as $2^n$ (because it grows by 4 every 2 steps; however, it still grows asymptotically slower) which is easily shown to be $ω(f_2)$.
How do I prove this relationship?

Express combinations number via factorials, and then use Stirling's Approximation for factorials. Then you'll be able to prove what you want.
