Motivated by the central limit theorem, one expects that
 $$\binom{n}{k} \approx \frac{2^n}{\sqrt{\pi n/2}} \exp\left(-\frac{(k-n/2)^2}{n/2}\right).$$
Computations suggest that the ratio of the two sides approaches 1 only for $|k-n/2| < 2\sqrt{n}$, and presumably this will follow from some version of the CLT.


In the literature or standard usage, are there any explicit upper (and lower?) bounds for binomial coefficients with a similar form that are sharp (in the ratio sense) for a wider range of $k$?



Are there good bounds on binomial coefficients?
Let $h(x)=-x\ln x-(1-x)\ln (1-x)$ be the binary entropy function in nats, then for $k\in [1,n-1]\cap \mathbb{Z}$ we have
$$
\sqrt{\frac{n}{8k(n-k)}}\exp\{nh(k/n)\} \leq \binom{n}{k} \leq \sqrt{\frac{n}{2\pi k(n-k)}}\exp\{nh(k/n)\}
$$
where the upper bound approaches equality if $k$ and $n-k$ are both large. This is obtained from Stirling and then some other manipulation, and covers the whole range of $k$.


This result is most certainly not mine, I learned it from Bob Gallager's *Information Theory and Reliable Communications*.


