Namely, given a sequence a0, a1, a2, ..., we can form its exponential generating function . Then the ordinary generating function is given by the integral
when this integral exists. (This is something like the Laplace transform.)
The proof is straightforward, and I'll give it here. (I'll ignore issues of convergence.) Consider that integral; by recalling what A(z) is, it can be rewritten as
and we can interchange the integral and the sum to get
But we can pull out the part of the integrand which doesn't depend on t to get
and that integral is well-known as the integral representation for the gamma function. The integral is k!, which cancels with the k! in the denominator, and we recover .
It's easy to come up with examples to verify this; for example, the exponential generating function of the binomial coefficients is z2ez/2; plugging this into our formula gives z2(1-z)-3, which is the ordinary generating function of the same sequence.
I'm not sure how useful it is to know this, though... I can't think of too many times where I had an exponential generating function for some sequence and thought "oh, if only I had the ordinary generating function."