<p>Different types of omnibus goodness-of-fit tests for a binomial null hypothesis are developed, which are based on the counts’ probability generating function. For each statistic, closed-form asymptotics are derived and used to implement the tests in practice without the need for a bootstrap procedure. The finite-sample performance of the tests is analyzed by means of a comprehensive simulation study, and practical recommendations for the choice of the test statistic are given. The practical application of the tests is illustrated by a couple of real-world data examples.</p>