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

By AJ Bu, Shalosh B. Ekhad, and Doron Zeilberger

.pdf    .tex   

Written: May 5, 2024

We show the utility of AJ Bu's recent article for computing explicit expressions for the GENERATING functions of sums of powers of areas under Dyck and Motzkin paths, by deducing 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.

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilerger, AJ Bu's web-site, and arxiv.org .

Maple package

Sample Input and Output for qEWplus.txt

  • If you want to see explicit expressions for the sum of the r-th powers of areas under Dyck walks from 0 to 2n for r from 1 to 20
    the input gives the output

  • If you want to see explicit expressions for the sum of the r-th powers of areas under Motzkin walks from 0 to n for r from 1 to 20
    the input gives the output

    Doron Zeilberger's Home Page

    AJ Bu's Home Page