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: Is there a better estimate for the constant $C$ than the one given above for the special case of the binomial distribution?
Reason for my question: The given inequality for $C$ holds for any standardized sum of any i.i.d random variables. But I am only interested in the case of binomial distributed random variables. From the answer to my question Estimates for the normal approximation of the binomial distribution I know, that I cannot except a better estimation. But I guess, that there is a better estimate of $C$ if one restricts the Berry-Esseen theorem to binomial distributions only. It would be great when you can point me to an article or a textbook with a better estimate of $C$.
Update: I reasked the question on math.stackexchange.com as suggested in the comments. I hope, this is okay.