Generating function

Generating function --- производящая функция.

A generating function of a sequence [math]\displaystyle{ a_{0}, a_{1}, \ldots, a_{n}, \ldots }[/math] is the function

[math]\displaystyle{ f(x) = \sum_{n=0}^{\infty} a_{n}x^{n}. }[/math]

Exponential generating function is the function

[math]\displaystyle{ f(x) = \sum_{n=0}^{\infty} \frac{a_{n}}{n!} x^{n}. }[/math]