The function c(x) = (1 - sqrt(1 - 4x))/(2x) is known as the generating function for the Catalan numbers, which are a sequence of natural numbers with numerous combinatorial applications. This generating function encapsulates the Catalan numbers in such a way that it allows us to derive their values through algebraic manipulation and limits. Catalan numbers are crucial in counting problems, such as the number of valid parentheses combinations or the number of paths along a grid that do not cross above a diagonal.
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