# Catalan and Motzkin Integral Representations

## TL;DR

This paper offers simplified proofs of integral representations for Catalan and Motzkin numbers using standard calculus techniques, expanding the mathematical understanding of these combinatorial sequences.

## Contribution

It introduces new, straightforward proofs for existing Catalan integral representations and develops analogous representations for Motzkin numbers.

## Key findings

- Eight new integral representations of Catalan numbers proved.
- New integral representations for Motzkin numbers established.
- Simplified proof techniques demonstrated for combinatorial sequences.

## Abstract

We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in the literature are proved by using advanced mathematical techniques. All integral representations in this paper are proved by using standard techniques from integral calculus. Thus, we provide a more simplistic approach of proving integral representations of the Catalan and Motzkin numbers.