Generating function

In mathematics, a generating function is a function for which the definition "encodes" values of a sequence, allowing the application of methods of real and complex analysis to problems in algorithmics, combinatorics, number theory, probability and other areas.

Let (a_n) be a sequence indexed by the natural numbers. The ordinary generating function may be defined purely formally as a power series

$A(z)=\sum _{n=0}^{\infty }a_{n}z^{n},\,$

where for the present we do not address issues of convergence.

The exponential generating function may be defined similarly as a power series

$A(z)=\sum _{n=0}^{\infty }{\frac {a_{n}}{n!}}z^{n}.\,$