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 $g_1(y)$.