Working with cellular automata I came across a system of equations for unknown integers $R_{k}$ and $C_{k}$ that looks like this.

$\binom{m}{k}=R_{k}+C_{k}+\sum\limits_{j=1}^{k-1}R_{j}C_{k-j}.$

Where 0< k$\leq$ 2m

(for k>m we take $\binom{m}{k}=R_{k}=C_{k}=0$)

Given $R_{1}$, the system has a unique solution.

Has anyone seen something similar? Do you know if it still possible to solve it, if instead of $\binom{m}{k}$ in the left you put something else?

I just want to know if it's related to something else, and if it's possible to solve more general systems.