Taylor Series / Power Series

Alright, I'm just a little confused on what my work is asking me. The problem says "Use the definition to find the Taylor series, centered at c, for the function."

The problem is:

So I have an answer book that has the answer for this problem. I understand Taylor series / power series, but I'm just a bit confused on what is going on.

The answer book goes on to do a few derivatives, plugs in the c, etc. Then it forms the sum which is:

Now I understand this is in the form of:

Yet... Why does it not follow the "basic list" of power series? Which has:

Why is there only n powers in the answer, and no 2n to be found, etc.? Can someone please explain? Maybe I am mixing two concepts...