  A goodness-of-fit test for one-parameter count distributions with finite
second moment is proposed. The test statistic is derived from the $L^1$
distance of a function of the probability generating function of the model
under the null hypothesis and that of the random variable actually generating
data, when the latter belongs to a suitable wide class of alternatives. The
test statistic has a rather simple form and it is asymptotically normally
distributed under the null hypothesis, allowing a straightforward
implementation of the test. Moreover, the test is consistent for alternative
distributions belonging to the class, but also for all the alternative
distributions whose probability of zero is different from that under the null
hypothesis. Thus, the use of the test is proposed and investigated also for
alternatives not in the class. The finite-sample properties of the test are
assessed by means of an extensive simulation study.
