  The study of Chemical Reaction Networks (CRN's) is a very active field.
Earlier well-known results \cite{Feinberg:def_01:87, Anderson:product_dist:10}
identify a topological quantity called deficiency, for any CRN, which, when
exactly equal to zero, leads to a unique factorized steady-state for these
networks. No results exist however for the steady states of non-zero-deficiency
networks. In this paper, we show how to write the full moment-hierarchy for any
non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for
the factorial moments (FM). Using these, we can recursively predict values for
lower moments from higher moments, reversing the procedure usually used to
solve moment hierarchies. We show, for non-trivial examples, that in this
manner we can predict to high accuracy, any moment of interest, for CRN's with
non-zero deficiency and non-factorizable steady states.
