Methods used to infer nuclear parameters from neutron count statistics fall into two categories depending on whether they use moments or count number probabilities. As probabilities are in general more difficult to calculate, we are interested here in the reconstruction of distributions from their lower-order moments. For this, we explore two approaches. The first one relies on a generalization of the two-forked branching correlation (quadratic) approximation used in the PMZBB and Poisson radical distributions, and the second one is founded on the expansion of the distribution on a Meixner discrete orthogonal polynomial base.