I want to solve recurrence relation using generating functions. One part is $$\sum_{n\geq 2}(n+1)z^n$$ I know that there is a generating function for $$\sum_{n\geq 0}(n+1)z^n$$ so I need to shift the index (I guess?) First I start with $$z^2\sum_{n\geq 2}(n+1)z^{(n-2)}$$
But how do I Index-Shift correctly?
My first idea would have been something like: $$z^2\bigg(\Big(\sum_{n\geq 0}(n+1)z^n\Big) -1-2z\bigg) = z^2\bigg(\frac{1}{(1-z)^2}-1-2z\bigg)$$ Substracting the parts for n=0 and n=1. But I have no idea if I can really do it like that.
Can anyone help me? (Confirm or show how to do it better)
The whole assignment is for: $a_n=2a_{n-1}-a_{n-2}+n+1, n\geq2, a_0=1,a_1=4$