Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form

Let $X\sim Bin(N,p)$ and let $c=Nr$ be a prescribed centre with $r\ne p$. The finite formula for $E|X-c|$ is classical, and the existence of a complete large-deviation expansion in powers of $N^{-1}$ follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in close