Asymptotic expansions for the distributions of sums of independent nonnegative integer random variables in the neighbourhood of the Poisson distribution are derived, together with explicit estimates of the truncation error. Expansions are also derived for the expectations of at most polynomially growing functions of such sums. Applications to the Poisson binomial and Poisson negative binomial approximations are considered. The method used is an adaptation of the Stein-Chen approach.
A. D. Barbour. "Asymptotic Expansions in the Poisson Limit Theorem." Ann. Probab. 15 (2) 748 - 766, April, 1987. https://doi.org/10.1214/aop/1176992169