A class of infinitely divisible distributions connected to branching processes and random walks

A class of infinitely divisible distributions on {0,1,2,…} is defined by requiring the (discrete) Lévy function to be equal to the probability function except for a very simple factor. These distributions turn out to be special cases of the total offspring distributions in (sub)critical branching processes and can also be interpreted as first passage times in certain random walks. There are connections with Lambert's W function and generalized negative binomial convolutions.