A generating function is a formal structure that is closely related to a numerical sequence, but allows us to manipulate the sequence as a single entity, with the goal of understanding it better. Roughly speaking, generating functions transform problems about sequences into problems about
functions. They provide a systematic way to encode sequences of numbers or other combinatorial objects, allowing for elegant solutions to complex problems across diverse mathematical domains.

In this article, we will approach a range of problems, involving placing $n$ chess pieces on an $n\times n$ chessboard so that no two pieces attack each other, using the generating functions approach.

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