Solve $a_{n} = a_{n-1} - c_{n-2} a_{n-2}$

Question

Is there a way to write a solution of the recurrence relation:

$$a_{n} = a_{n-1} - c_{n-2} a_{n-2}$$

with the initial conditions $a_0=1$ and some fixed positive value for $a_1$. Here the $c_n$ are positive numbers.