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 $k$?