I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of
$$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right|$$
From the Berry-Essen theorem I can deduce
$$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$
with $C \le 0.4748$.
My question: Are there better estimates for the normal approximation of the binomial distribution?
The Berry-Esseen theorem is quite general because it can be applied to each sum of i.i.d random variables. So I guess there are better estimates for the special case of the binomial distribution...