A generating function is a formal power series used in combinatorics to encode sequences of numbers. It transforms a sequence into a function, allowing mathematicians to manipulate and analyze the sequence more easily. For example, the generating function for the sequence of natural numbers can be expressed as a power series, where the coefficients represent the terms of the sequence. Generating functions can be classified into different types, such as ordinary generating functions and exponential generating functions. They are useful for solving problems related to counting, recurrence relations, and probability. By using generating functions, one can derive formulas and find closed forms for sequences, making them a powerful tool in combinatorial mathematics.