Absolute moments of the binomial distribution folded at its mean

Source abstract

Let $X\sim Bin(N,p)$ and $Y=|X-Np|$. We derive an exact reduction for every odd absolute moment of $Y$, expressing it through finitely many central masses and central tail probabilities. The tail coefficients satisfy a sum rule and are divisible by $q-p$, so that in the symmetric case the odd ladder collapses to masses alone. We then obtain the complete central-tail expansion at a bounded lattice displacement, with Bernoulli-polynomial coefficients in integer powers of the large parameter. As an application, the first two tail coefficients yield a second-order expansion for the median of the beta distribution, whose formal one-parameter degeneration reproduces the first two terms of Choi's expansion for the gamma median.