  Random walks are a series of up, down, and level steps that enumerate
distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the
path. We used these paths to analyze Catalan, Schr\"{o}der, and Motzkin number
sequences through a combination of matrix operations, quadratic functions, and
inductive reasoning. Our results revealed a number of distinct patterns, some
unnamed, between these number sequences and Pascal's triangle that can be
explained through generating functions, first returns, group theory, and the
Riordan matrix.
  Various proofs and properties of these number sequences are provided,
including each generating function, their respective first returns, and matrix
properties. These findings lead to a deeper understanding of combinatorics and
graph theory.
