I'm not sure what you want me to check.

These 0,1 rvs are called Bernoullis and their sum is a Binomial.

You have a few typos.

C runs from 1 to n.

You accidentally are starting at 0.

O,1,...,n are n+1 terms.

But n/2 is correct.

You want 1,...,n in your sums.

Also that is just weird.

It can't be a lower value since it's greater than n, your upper limit.

You want 1,..,n in your series, all of them.