Solving $a_n = c_{n,n - 1} a_{n - 1} + \dots + c_{n,0} a_{0}$

Question

I have been drawn to the art of solving recurrence relations from an unrelated problem in analysis. Given real numbers $a_0$ and \begin{eqnarray} & & c_{1,0} \\ & c_{2,1} \quad & c_{2, 0} \\ c_{3,2} \quad & c_{3,1} \quad & c_{3,0} \\ & & \vdots \quad, \end{eqnarray} is there a way to explicitly solve $a_n = c_{n,n - 1} a_{n - 1} + \dots + c_{n,0} a_{0}$? One can work out the first few terms: $$ a_1 = c_{1,0} a_0 \quad; \quad a_2 = c_{2,1} a_1 + c_{2, 0} a_0 = \left(c_{2,1} c_{1,0} + c_{2, 0}\right) a_0 \quad ; \text{ etc.} $$ However, I can't figure out how to extract a closed formula for $a_n$. I have gone through a bit of literature but it seems that not many results can be applied to this problem.