Explicit generating functions for the sum of the areas under Dyck and Motzkin paths (and for their powers)

This paper first describes how to find the generating function for the sum of the areas under generalized Dyck paths (with an arbitrary set of steps), using Motzkin paths as a motivating example.Then, it focuses on deriving functional equations for the Motzkin and Dyck paths.Subsequently, it describes an algorithm to manipulate these functional equations for finding 'perturbation expansions' of the solutions, with applications to derive explicit generating functions for the sum of powers of areas under Dyck and Motzkin paths for any desired power.