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Generating function in mathematics is a way to encode an infinite sequence of numbers as a formal power series. The coefficients of the series correspond to the terms of the sequence. This technique is particularly useful in combinatorics for solving recurrence relations and counting problems.

A generating function $G(x)$ for a sequence $a_0, a_1, a_2, \dots$ is typically defined as the power series $G(x) = a_0 + a_1x + a_2x^2 + \dots = \sum_{n=0}^\infty a_n x^n$. While this is a formal power series, it can also be viewed as a function in a region of convergence for the series.

The power of generating functions lies in their ability to transform problems about sequences into problems about functions. For example, operations on sequences, such as addition, convolution, and differentiation, translate into algebraic operations on their corresponding generating functions. This can simplify complex combinatorial identities and recurrence relations, making them easier to solve or prove.

A common application is in solving linear recurrence relations. If a sequence is defined by a linear recurrence, its generating function can often be expressed as a rational function, which can then be expanded back into the sequence's terms. This provides a closed-form expression for the sequence.

Another key use is in counting problems. For instance, the number of ways to form a sum using a set of integers can be represented by a generating function where the coefficient of $x^k$ is the number of ways to obtain the sum $k$. The product of such generating functions can then be used to count combinations of these sums.