<s>[INST] How is ${n \choose n/2}$ approximated in this manner? In this question, the author asked how to get an asymptotic growth of ${n \choose n/2}$, to which the answer is as follows. First:
$$ {n \choose n/2} = \frac{n!}{(n/2)!(n- n/2)!} = \frac{n!}{(\tfrac{n}{2}!)^2} $$
Now, to find an upper bound on this, take the upper bound of Stirling's approximation for $n!$ and the lower bound for $\tfrac{n}{2}!$ which if I'm not mistaking gives the following:
$$ {n \choose n/2} \leq \frac{\sqrt{2 \pi n} (\frac{n}{e})^n e^{\tfrac{1}{12n}}}{\Big( \sqrt{2 \pi \tfrac{n}{2}} (\frac{\tfrac{n}{2}}{e})^\tfrac{n}{2} e^{\tfrac{1}{12\tfrac{n}{2}+1}}\Big )^2} $$
However, the answer states this should give:
$$ {n \choose n/2} \leq \frac{e \cdot n^{n + \tfrac{1}{2}} e^{-n}}{\Big( \sqrt{2 \pi} \cdot \tfrac{n}{2}^\tfrac{n+1}{2} e^{-\tfrac{n}{2}}\Big )^2} $$
I'm confused as to how one would get from the former to the latter. Up to now I get stuck around here:
$$ {n \choose n/2} \leq \frac{\sqrt{2 \pi n} (\frac{n}{e})^n e^{\tfrac{1}{12n}}}
{\Big( \sqrt{2 \pi \tfrac{n}{2}} (\frac{\tfrac{n}{2}}{e})^\tfrac{n}{2} e^{\tfrac{1}{12\tfrac{n}{2}+1}}\Big )^2} \\
\leq \frac{\sqrt{2 \pi n} \cdot n^n e^{-n} e^{\tfrac{1}{12n}}}
{\Big( \sqrt{2 \pi \tfrac{n}{2}} \cdot \tfrac{n}{2}^\tfrac{n}{2} e^{-\tfrac{n}{2}} e^{\tfrac{1}{12\tfrac{n}{2}+1}}\Big )^2} \\
\leq \frac{\sqrt{2 \pi} \cdot n^{n + \tfrac{1}{2}} e^{-n} e^{\tfrac{1}{12n}}}
{\Big( \sqrt{2 \pi} \cdot \tfrac{n}{2}^\tfrac{n+1}{2} e^{-\tfrac{n}{2}} e^{\tfrac{1}{12\tfrac{n}{2}+1}}\Big )^2}
$$
Now, do the latter terms just disappear due to $\lim_{n \rightarrow \infty} \frac{e^{\tfrac{1}{12n}}}{\Big ( e^{\tfrac{1}{12\tfrac{n}{2}+1}}\Big )^2} = 0$? But in this case, why should there still be an $e$ term in the numerator? Also what happened to the $\sqrt{2 \pi}$ in the numerator? It cannot have canceled out with the one in the denominator since the latter is still there...
 [/INST] Hint: It seems the answerer didn't use the inequality chain stated in the Wiki-page
as starting point, but started instead with the asymptotic formula
\begin{align*}
\color{blue}{n!\sim\left(\frac{n}{e}\right)^n\sqrt{2\pi n}}\tag{1}
\end{align*}
We obtain from (1) assuming even $n$:
\begin{align*}
\color{blue}{\left(\frac{n}{2}\right)!}&\sim\left(\frac{n/2}{e}\right)^{n/2}\sqrt{\pi n}=\left(\frac{n}{2e}\right)^{n/2}\sqrt{\pi n}\\
&=\left(\frac{n}{2}\right)^{n/2}e^{-n/2}\sqrt{\pi n}=\left(\frac{n}{2}\right)^{n/2}e^{-n/2}\sqrt{2\pi}\sqrt{\frac{n}{2}}\\
&\color{blue}{=\left(\frac{n}{2}\right)^{\frac{n+1}{2}}\sqrt{2\pi}e^{-n/2}}\tag{2}
\end{align*}
and squaring (2) gives the denominator of the expression under consideration
\begin{align*}
\binom{n}{n/2}\stackrel{?}{\leq}\frac{e\,n^{n+\frac{1}{2}}e^{-n}}{\left(\left(\frac{n}{2}\right)^{\frac{n+1}{2}}\sqrt{2\pi}e^{-n/2}\right)^2}
\end{align*}
From (1) and (2) we derive the asymptotic formula
\begin{align*}
\color{blue}{\binom{n}{n/2}}&\sim\frac{\left(\frac{n}{e}\right)^n\sqrt{2\pi n}}{\left(\left(\frac{n}{2}\right)^{\frac{n+1}{2}}\sqrt{2\pi}\,e^{-n/2}\right)^2}\\
&=\frac{\sqrt{2\pi}\,n^{n+\frac{1}{2}}\,e^{-n}}{\left(\left(\frac{n}{2}\right)^{\frac{n+1}{2}}\sqrt{2\pi}\,e^{-n/2}\right)^2}\tag{$ \sqrt{2\pi}\leq e$}\\
&\,\,\color{blue}{\leq \frac{e\,n^{n+\frac{1}{2}}\,e^{-n}}{\left(\left(\frac{n}{2}\right)^{\frac{n+1}{2}}\sqrt{2\pi}\,e^{-n/2}\right)^2}}
\end{align*}
where the last inequality holds only if $n$ is sufficiently large.
</s>