Geodesic
Some explicit formulas in the theory of branching stochastic processes with discrete time and one-type particles
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 2, pp. 348-354 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $Z_0=1$, $Z_n$, $n=1,2,\dots,$ be the number of particles belonging to the $n$-th generation of a Halton–Watson branching process, $p_{nr}=\mathbf P\{Z_n=r\}$ and $F_n(z)=\sum_{r\ge0}p_{nr}z^r$. It is supposed that $m=\mathbf EZ_1\ne1$ and $F(z)=F_1(z)$ be regular at the point $z=1$ for $m<1$. In the paper some formulas and an asymptotic expansion for the probabilities $p_{nr}$ are obtained. 
@article{TVP_1969_14_2_a16,
     author = {B. I. Selivanov},
     title = {Some explicit formulas in the theory of branching stochastic processes with discrete time and one-type particles},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {348--354},
     year = {1969},
     volume = {14},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_2_a16/}
}
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B. I. Selivanov. Some explicit formulas in the theory of branching stochastic processes with discrete time and one-type particles. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 2, pp. 348-354. http://geodesic.mathdoc.fr/item/TVP_1969_14_2_a16/