  Narayana numbers appear in many places in combinatorics and probability, and
it is known that they are asymptotically normal. Using Stein's method of
exchangeable pairs, we provide an error of approximation in total variation to
a symmetric binomial distribution of order~$n^{-1}$, which also implies a
Kolmogorov bound of order~$n^{-1/2}$ for the normal approximation. Our
exchangeable pair is based on a birth-death chain and has remarkable
properties, which allow us to perform some otherwise tricky moment
computations. Although our main interest is in Narayana numbers, we show that
our main abstract result can also give improved convergence rates for the
Poisson-binomial and the hypergeometric distributions.
