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

Lattice paths with a first return decomposition constrained by the maximal height of a pattern