Lattice Paths With a First Return Decomposition Constrained by the Maximal Height of a Pattern

We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form $A(x)=\sum_{k\geq 0}A_k(x)$. apply this general result enumeration certain subsets Dyck, Motzkin, skew Motzkin paths, defined recursively according first return decomposition with monotonically non-increasing condition relative maximal ordinate reached by an occurrence pattern $\pi$.