recurrence and generating functions

Suppose with .

Then,

.

So,

Solving for in

we obtain

and

Thus,

.

So, the general formula for the recurrence is given by

.

According to my exam, the correct answer is , which I get when I use the method of undetermined coefficients. However, I am not seeing my mistake when I use the method above.

Also, lets say I have a recurrence relation starting with intitial contition . Do I just start the infinite series at one and move on like any other problem (i.e., )? Finally, under what conditions does this method for solving recurrence relations fail?