Let $\{Z_t\}_{t\in\mathbb N}$ a branching process with one immigrant. We suppose that knowing $Z_0,...,Z_t$ the r.v. $Z_{t+1}$ is a sum of $Z_t+1$ r.v. which take value $k$ with probability $\alpha _t(1-\alpha _t)^k$ whith $\alpha _t\in (0,1)$ for all $t$. Let $Z_{s,t}$ the number of progeny alive at time $t$ of the immigrant who entered at time $s<t$. We can use the following representation of $Z_t$ :$$Z_t=\sum_{s=0}^{t-1}Z_{s,t}.$$
How can I compute $\mathbb E[Z_{s,t}]$ ? I know that $$\mathbb E[Z_{s,t}]=\prod_{i=0}^{t-1}\frac{1-\alpha _i}{\alpha _i},$$
but I don't see how to prove it.
I know that $Z _{t}=\sum_{i=0}^{Z_{t-1}}X_i$ where $X_i\sim Geom(\alpha_t),$ and thus I know $$\mathbb E[Z_t\mid Z_{t-1}]=(Z_{t-1}+1)\sum_{k=0}^\infty \alpha _t(1-\alpha _t)^k.$$
But I can't get $\mathbb E[Z_{s,t}]$.