Finding Recurrence Relation in Closed Form or as Infinite Summation/Coproduct

Question

My recurrence relation is:

$$f(n)=(2n-3) f(n-1) -x^2 f(n-2)$$

Where

$$f(-1)=1$$ $$f(0)=0$$ $$f(1)=-x^2$$ $$f(2)=f(1)$$ $$f(3)=-x^2 (3-x^2)$$

And it gets more complicated from there