I have a Galton Watson process where the offspring distribution is $P(X=k) = (1-p)^kp$ for $k=0,1,2,\ldots$. Let $T=t$ be the event that the population (denote it $Z_0,Z_1,Z_2,\ldots$ with $Z_0 = 1$) is first empty at time $t$. I want to find $P(T=t)$.
The generating function of $X$ is $$G_X(s) = \frac{p}{1-s(1-p)}.$$ Define $G^1(s) = G_X(s)$ and $G^{i+1}(s) =G(G^i(s))$. Then I have managed to show that the answer should be \begin{equation} G^t(0) - G^{t-1}(0). \end{equation} Now, $G^1(0) = p$ and $G^2(0) = p/(1-p(1-p))$. But after this I cannot find any reasonable expresssion for the solution... Did I make a mistake somewhere