TITLE: Convert an OGF to an EGF
QUESTION [7 upvotes]: If I have an ordinary generating function (OGF) as a rational polynomial:
$$
Q(y) = \frac{f_1(y)}{g_1(y)}
$$
Which has a power series representation 
$$
Q(y) = \sum_{n \geq 0} a_n y^n
$$
How can I compute the exponential generating function (EGF)
$$
Z(y) = \sum_{n \geq 0} b_n y^n / n!
$$
as a rational polynomial?
$$
Z(y) = \frac{f_2(y)}{g_2(y)}
$$
As an explicit solved example, consider the case where
$$ Q(y) = \frac{yv -1}{y(2+v) -1} $$
The intermediate terms are
$$ a_n  = 2 (v+2) ^ {n-1} $$
Giving as a final answer
$$ Z(y) = \frac{2 e^{(v+2)y}}{v+2} $$
... ideally, I like to be able to do this by going from one rational polynomial to another without computing $a_n$. We can assume that I can factor the polynomial $g_1(y)$.

REPLY [6 votes]: What you want is the Borel transform.  This is related to the inverse Laplace transform.  Specifically, $Z(s)$ is the inverse Laplace transform of $Q(1/t)/t$.