How can I solve the recurrence relation below using generating functions? $$f(n) = \frac{9}{2}f(n-1) + \frac{9}{2}f(n-2) - \frac{81}{2}f(n-3) + \frac{81}{2}f(n-4) + n + 2$$ $$f(0) = f(1) = f(2) = f(3) = 1$$ Attempt:
The generating function of (n+2) is $\frac{2-x}{(1-x)^2}$. So we can write our generating function this way, $f = \frac{9}{2}xf + \frac{9}{2}x^2f - \frac{81}{2}x^3f + \frac{81}{2}x^4f + \frac{2-x}{(1-x)^2}$.
Let's isolate f and get, $f(1 - \frac{9}{2}x - \frac{9}{2}x^2 + \frac{81}{2}x^3 - \frac{81}{2}x^4) = \frac{2-x}{(1-x)^2}$ and finally, $f = \frac{2-x}{(1 - \frac{9}{2}x - \frac{9}{2}x^2 + \frac{81}{2}x^3 - \frac{81}{2}x^4)(1-x)^2}$.
Here I got stuck.