Generating function

A generating function is a mathematical tool that encodes a sequence of numbers into a single algebraic expression, typically a power series. It acts like a compact summary, allowing us to analyze and manipulate the entire sequence through algebraic operations. Think of it as a "machine" where each coefficient corresponds to a term in the sequence, making it easier to find patterns, sums, or indices without handling each term individually. Generating functions are commonly used in combinatorics, probability, and other fields to solve problems involving sequences and their properties efficiently.