Is there such a thing as (-2)! or (-1)!?
I want to find the generating function for a_{n}=n(n-1).
Am I justified in the second to last step, ignoring the summands for k=-1 and k=-2?
If so, is there a reason why I can ignore the first two terms?
Only in this case, because you can see if you plug n=0 and n=1 into your original sum that those terms are 0 anyway.
Really what you should have written is
It doesn't make sense to put into your counter values which will give undefined terms.