Generating functions are mathematical tools used to encode sequences of numbers as coefficients in a power series. They provide a way to study sequences by transforming them into functions, making it easier to manipulate and analyze them. For example, the generating function for the sequence of natural numbers can be expressed as a power series, allowing for the exploration of properties like sums and recurrences. These functions are particularly useful in combinatorics, where they help solve counting problems and find closed forms for sequences. By using generating functions, mathematicians can derive relationships between different sequences and uncover deeper insights into their behavior, such as those found in Fibonacci numbers or binomial coefficients.