Generating functions

Generating functions are mathematical tools that encode sequences—like numbers arranged in order—into a single algebraic expression, usually a power series. Think of it as a compact way to represent an entire sequence, where each term's position corresponds to a specific power of a variable (often \(x\)). This approach allows mathematicians to analyze and manipulate sequences more easily, solving problems related to counting, patterns, or recursive relations by working with the generating function instead of individual terms. In essence, generating functions translate sequences into a form suitable for algebraic techniques and problem-solving.