Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers)

In this paper, we first describe 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. We then focus on Motzkin and Dyck paths, deriving functional equations for them. We then describe an algorithm to manipulate these functional equations for finding `perturbation expansions’ of the solutions, with applications for deriving explicit generating functions for the sum of powers of areas under Dyck and Motzkin paths for any desired power.