Generating functions are mathematical tools used to encode sequences of numbers as coefficients in a power series. They provide a way to manipulate and analyze sequences, making it easier to solve problems in combinatorics and number theory. For example, the generating function for the sequence of natural numbers can be expressed as a power series, where each term corresponds to a number in the sequence. These functions can be classified into different types, such as ordinary generating functions and exponential generating functions, depending on the context. They are particularly useful for finding closed-form expressions for sequences and for solving recurrence relations, which are equations that define sequences recursively.