In recent years, there has been growing interest in shape-restricted inference. One such assumption is log-concavity of the distribution. Even when this assumption is not valid, the resulting maximum likelihood estimate still possesses desirable properties. In this work, we apply log-concave probability mass functions (pmfs) to time-series of counts. For continuous data, the log-concave fit is often smoothed with a Gaussian density to overcome the limitations of the empirical estimate; namely, its lack of smoothness and finite support. While smoothness is not a concern for count data, finite support is certainly an issue. In this work, we smooth the estimate with the Skellam distribution, resulting in an estimate that remains log-concave and has desirable properties. We then apply this idea to time series of counts, where the data given a mean process is assumed to be log-concave. The mean process, which determines the temporal dependence, follows a recursion based on previous observations and mean values. Smoothing is the final step in the estimation of the conditional pmf. The results are illustrated with simulations and real data. (This is joint work with Richard Davis).