Consider a country of $N$ people, where a president is elected every year. He is picked at random and his mandate can be renewed any number of times, but once he steps off he can't be re-elected.
Call $p_t(n)$ the probability that at time $t$ there are $n$ viable candidates, besides the current president. We have
$$p_{t+1}(n) = \frac{n+1}{n+2} p_t(n+1) + \frac{1}{n+1} p_t(n)$$
with the initial condition $p_0(n)=\delta_{nN-1}$.
How would you tackle such a recurrence equation?
In terms of the generating functional $F_z(n) = \sum_{t\geq 0} P_t(n) z^t$, I can write
$$\frac{n+1}{n+2} F_z(n+1) = \left(\frac{1}{z} - \frac{1}{n+1}\right) F_z(n) - \frac{\delta_{n N-1}}{z}$$
but I'm not sure how to proceed then.