  Operators on probability distributions can be expressed as operators on the
associated moment sequences, and so correspond to operators on integer
sequences. Thus, there is an opportunity to apply each theory to the other.
Moreover, probability models can be sources of integer sequences, both
classical and new, as we show by considering the classical M/G/1 single-server
queueing model. We identify moment sequences that are integer sequences. We
establish connections between the M/M/1 busy period distribution and the
Catalan and Schroeder numbers.
