Recurrence relation problem

Hi I'm studying for my finals and having trouble with this question:

*Consider the following recurrence relation:*

for where .

Write the corresponding generating function in its closed form.

I've set up the generating function as

and I've gotten as far in my working as:

But according to the solutions I should at this stage be getting:

What am I doing wrong?

I'd really appreciate some help on this one, thanks in advance!

Cheers,

Roro

Re: Recurrence relation problem

We have the following table where I called the function in the last row F(z):

In the right part of the table (after the vertical line), the first row equals the sum of the other rows. Using , this gives