Explore Euler's triangulation problem and the fascinating Catalan numbers in this mathematics lecture. Learn about Euler's formula for counting polygon triangulations, derive the formula for Catalan numbers, and discover their various applications in combinatorics. Investigate the sequence's connection to Newton's binomial expansion for sqrt(1-4x) and engage with smaller problems presented throughout the lecture. Gain insights into this important mathematical concept and its diverse manifestations in different areas of mathematics.