The paper studies different moment characteristics of (discrete) distributions: general, central, binomial, and factorial moments, as well as ordinary and factorial semi-invariants. There are established connections between various kinds of moments and recomputation formulas of arbitrary order. Numerical and analytical recursive relations that express the moments of higher order in terms of lower-order moments are provided. This problem appears to be closely related to combinatorics, to polynomials and numbers, and the corresponding formulas and relations for the moment characteristics are to a large extent merely consequences of the properties of Bell polynomials, Stirling numbers, etc.