generating function

A generating function is a mathematical object that represents an infinite sequence of numbers $(c_{n})_{n \geq 0}$ as the coefficients of a formal power series (formal as in we don't care about convergence).

f (x) = \sum_{n = 0}^{\infty} c_{n} x^{n}

To get the coefficient of $x^{n}$ , we use the notation

[x^{n}] f (x) = c_{n}

Examples

Properties

$[x^{n}] (f \cdot f) = \sum_{k = 1}^{n} [x^{k}] f \cdot [x^{n - k}] f$
$[x^{n}] (x f) = [x^{n - 1}] f$ for $n \geq 1$ (and for $n = 0$ , it is 0)

generating function

generating function

Examples

Properties

References

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