another question on indices on generating functions

**Use the recurrence ** **to find the exponential generating function for .**

Here is my solution:

But when I looked at the solution, the was unnecessary.

This happened because I began the summation from 2.

I did this because (i) the problem stated , and (ii) wouldn't make sense.

Was this unnecessary?

Also, if the problem asked me to find "the generating function for ", does that mean

or

Finally, if there are some advices to not to get confused with indices (some general rules for handling them or something), I would much appreciate them.