Limits of areas under lattice paths

We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of the paths tend to infinity. We find the limiting series for generalized Schröder, Motzkin, and Catalan paths. The limiting series for Schröder paths and their generalizations are shown to count partitions with restrictions on the multiplicities of odd parts and no restrictions on even parts. The limiting series for generalized Motzkin and Catalan paths are shown to count generalized Frobenius partitions and some related arrays.