I have a branching process of the form $p_0=0.1$, $p_1 = 0.6$, $p_2 = 0.3$. (any other $p_n = 0$). $Z_0$, the original population is $1$. $Z_1$ is the population after $1$ timestep, $Z_2$ is the population after $2$ timesteps, etc.
How can I compute the probability mass function of $Z_3$? It would suffice to have an expression I could expand on my TI-89.