I was given a problem to simplify the following:

where:

whenever j > n.

It's easy to see that the problem is equivalent to:

Now I can see that for odd values of n, this expands to:

and since:

we can use the last n/2 terms to "fill in the gaps", simplifying it to be:

That's easy. But I don't know why the proof still works for even numbers of n. You no longer have the same correspondence (n choose 0 no longer has the equivalent n choose n-1 as a term):

So it can't "fill in the gaps" in the same way.

Can anyone figure this out? I've been at this for hours and the explanations I'm being given by the texts are just crap.