In this paper we characterize all the $r$-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the $r$th-order univariate Hermite distributions. Among these, we find the Poisson $(r=1)$ and the ordinary Hermite distributions $(r=2)$.
Pedro Puig. Jordi Valero. "Characterization of count data distributions involving additivity and binomial subsampling." Bernoulli 13 (2) 544 - 555, May 2007. https://doi.org/10.3150/07-BEJ6021