Is this a studied recurrence?

Usually generating functions are the best way to deal with this kind of recurrences.

Write $C(T) = \sum_{k \geq 0} c_kT^k$ and $W(T) = \sum_{k \geq 1} w_kT^{k - 1}$. The recurrence relations can then be packed into a differential equation of formal power series: $C' = -WC$.

This obviously has solution $C = \exp(-\int W)$ where $\int W$ is the formal integral of $W$, namely $\int W = \sum_{k \geq 1} \frac{w_k}k T^k$.

Thus we have $\sum_{k \geq 0} c_k T^k = \exp(-\sum_{k \geq 1} \frac{w_k}k T^k)$. This kind of constructions appear e.g. in the definition of zeta functions of algebraic varieties over finite fields: see e.g. local zeta function.

As you are not presenting an explicit question, I hope this information is enough to point you to eventually useful references.