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In this paper, we describe how to find the generating function for the sum ofthe areas under Motzkin paths of length $n$ through a method that usesdynamical programming, which we show can be expanded for paths with any givenset of steps that start and end at height zero and never have a negativeheight. We then explore the case where, instead of a set of steps, we are giventhe quadratic functional equation$f(x,q)=P(x,q)+Q(x,q)f(x,q)+R(x,q)f(x,q)f(qx,x)$. We present a fully automatedmethod for finding perturbation expansions of the solutions $f(x,q)$ to suchquadratic functional equations and demonstrate this method using Motzkin paths.More importantly, we combine computer algebra with calculus to automaticallyfind $\frac{d^k}{dq^k}\left[f(x,q) \right]\big|_{q=1}$, explicitly expressed interms of radicals. We use Dyck and Motzkin paths to exemplify how this can beused to find explicit generating functions for the sum of the areas under suchpaths and for the sum of a given power of the areas.

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