Explicit Expressions for the First 20 Moments of the Area Under Dyck and Motzkin Paths

Starting from AJ Bu’s recent article that computed explicit expressions for the GENERATING functions of sums of powers of areas under Dyck and Motzkin paths, we deduce from them explicit expressions for the actual sequences. This enables taking the limits of the scaled moments and confirming, in an entirely elementary way, that they tend to those of the area under Brownian Excursion (up to any specified moment).