<s>[INST] Understanding $n \left(\frac{2n \choose n}{4^n}\right)^2$ for large $n$ My answer over at cstheory.stackexchange.com involved the expression $$\lim_{n\to \infty} n \left(\frac{2n \choose n}{4^n}\right)^2$$ According to Wolfram Alpha, this expression is at most $\frac{1}{\pi}$.
I would like to know why this is the case, but I don't know enough about these kinds of expressions to figure this out myself.
 [/INST] Use Stirling's formula:
$$ n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n. $$
Using this formula, we get that
$$ \binom{2n}{n} \sim \frac{\sqrt{4\pi n} (2n/e)^{2n}}{2\pi n (n/e)^{2n}} = \frac{4^n}{\sqrt{\pi n}}, $$
and so
$$ n \left( \frac{\binom{2n}{n}}{4^n} \right)^2 \sim \frac{1}{\pi}. $$
There are variants of Stirling's formula with bounds on the error terms, for example
$$ n! = \sqrt{2\pi n} \left(\frac{n}{e}\right)^n e^{\Theta(1/n)} $$
(Here $\Theta(1/n)$ is some quantity in the range $[-C/n,C/n]$ for some absolute constant $C > 0$.)
Using this, we get
$$ \binom{2n}{n} = \frac{\sqrt{4\pi n} (2n/e)^{2n} e^{\Theta(1/n)}}{2\pi n (n/e)^{2n} e^{\Theta(1/n)}} = \frac{4^n}{\sqrt{\pi n}} e^{\Theta(1/n)}, $$
and so
$$ n \left( \frac{\binom{2n}{n}}{4^n} \right)^2 = \frac{e^{\Theta(1/n)}}{\pi} = \frac{1}{\pi} + \Theta\left(\frac{1}{n}\right). $$
For even more terms, check the Wikipedia article.
Your experiments seem to imply that the sequence $f(n) = n (\binom{2n}{n}/4^n)^2$ is increasing. To check this, let's compute the ratio of two adjacent terms:
$$
\begin{align*}
\frac{f(n+1)}{f(n)} &= \frac{n+1}{n} \frac{(2n+2)!^2 n!^4}{(2n)!^2 (n+1)!^4} \frac{4^{2n}}{4^{2n+2}} \\ &= \frac{n+1}{n} \frac{(2n+2)^2(2n+1)^2}{(n+1)^4} \frac{1}{16} \\ &=
\frac{(2n+1)^2}{4n(n+1)} = \frac{4n^2 + 4n + 1}{4n^2 + 4n} > 1.
\end{align*}
$$
This shows that $f(n) < 1/\pi$ and $f(n) \to 1/\pi$. Since $f(1) = 1/4$, we get the following explicit formula:
$$ f(n) = \frac{1}{4} \prod_{k=1}^{n-1} \frac{4k^2+4k+1}{4k^2+4k}. $$
This also shows that
$$ \pi = 4 \prod_{k=1}^\infty \frac{4k^2+4k}{4k^2+4k+1}. $$
Therefore
$$
\begin{align*}
f(n) &= \frac{1}{\pi} \prod_{k=n}^\infty \frac{4k^2+4k}{4k^2+4k+1} \\ &=
\frac{1}{\pi} \prod_{k=n}^\infty \left(1 - \Theta\left(\frac{1}{k^2}\right)\right) \\ &=
\frac{1}{\pi} \left(1 - \sum_{k=n}^\infty \Theta\left(\frac{1}{k^2}\right)\right) \\ &=
\frac{1}{\pi} \left(1 - \Theta\left(\frac{1}{n}\right)\right) = \frac{1}{\pi} - \Theta\left(\frac{1}{n}\right),
\end{align*}
$$
as we have obtained before. (Here $\Theta(1/n)$ is some quantity in the range $[C_0/n,C_1/n]$ for some positive constants $0 < C_0 < C_1$.) If we sweat more, we can get better estimates on the error from this formula for $f(n)$; this is left to the reader.
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