Well, the denominators look something like a factorial (except for that pesky 3 in the squared term; perhaps you could massage it to look better). Maybe you could set up a recurrence relation to figure out the numerators?
Aha. So it is a nice little factorial in the denominator. Well, the recurrence relation idea sounds nice, but I'm not sure how it would work in principle. Why don't you show all the details of the computations for those first few terms? Maybe something'll pop out.
If exists, then does as well, and they are equal. Since the first expression exists, you can use it to find the radius of convergence, which will work out to be 1.
If you think to calculate from the terms you listed out then it becomes pretty easy to figure out the form of the , though this isn't exactly a rigorous solution.