  The generalized Poisson distribution is well known to be a compound Poisson
distribution with Borel summands. As a generalization we present closed
formulas for compound Bartlett and Delaporte distributions with Borel summands
and a recursive structure for certain compound shifted Delaporte mixtures with
Borel summands. Our models are introduced in an actuarial context as claim
number distributions and are derived only with probabilistic arguments and
elementary combinatorial identities. In the actuarial context related compound
distributions are of importance as models for the total size of insurance
claims for which we present simple recursion formulas of Panjer type.
