The Galton-Watson process with mean one

Let Z n be the number of individuals in the n -th generation of a simple (discrete, one-type) Galton-Watson process, descended from a single ancestor. Write and assume, as usual, that 0 < F (0) < 1. It is well known that the p.g.f. of Z n is F n ( s ), the n -th functional iterate of F ( s ), and that if the process is construed as a Markov chain on the states 0, 1, 2,…, then its n -step transition probabilities are given by

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