On finding a recurrence relation, probably you want to start with
See what happens when you multiply everything out-- it should give you information about the coefficients of H(x).
A recurrence relation is not the same thing as finding an explicit expression for the coefficients. For example, if I write
then that's a recurrence relation. If I write
that's an explicit expression.
It's in finding an explicit expression for that you may find partial fractions useful.