asymptotics for binomials

Question

What is a published reference for the asymptotic equivalent of $ n \choose k$ with $k$ linear in $n$? I want both the entropy function and the denominator in $\sqrt{n}.$

Answer 1

These lecture notes by David Galvin state the asymptotic expression (see the beginning of Section 3.1): \begin{align*} \binom{n}{\alpha n} &= \frac{2^{H(\alpha)n}}{\sqrt{2\pi\, n\, \alpha(1-\alpha)}} [1+o(1)] \qquad\text{as $n\to\infty$} \end{align*} for $0<\alpha<1$. He does not provide a proof, only that it follows from Stirling's approximation, so I assume the proof must be straightforward.

Answer 2

The following appears (with proof) in Ash, Information Theory, 1965 as Lemma 4.7.1:

$$ \frac{2^{nH(p,1-p)}}{\sqrt{8np(1-p)}} \le {n \choose np} \le \frac{2^{nH(p,1-p)}}{\sqrt{2 \pi np(1-p)}} \,, $$ Here $H(p,1-p):=-p \log_2 p - (1-p) \log_2 (1-p)$ is the binary entropy function.