Your notation is slightly confusing. The title of your post "Binomial Expansion" could suggest two possible ways to interpret your notation. You might be using , which is a binomial coefficient, or you may be asking that we help you expand the following using Newton's Generalized Binomial Formula:
I believe you mean the former, so I will change the notation slightly:
where currently, we assume is only defined when are nonnegative integers. Then
So, (as you wrote in your post).
However, let's manipulate this a little before we subtract . Using Pascal's Rule, which is , we get:
Subtracting from both sides gives:
Let's look at this for small values of .
The pattern is that when is even, the first terms cancel each other out. When is odd, the first terms should add to .
To see that this is true, suppose is even. Then we have:
When is odd, then is even and is odd, so . When is even, then is odd and is even, so . Hence, the only term that remains is the .
See if you can figure out how to show that when is odd, you get .