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.