Given the set of all binary strings of length n, I am looking at the "middle" of these strings, weight-wise.

Namely, I am trying to calculate how many words are there whose weight is between n/2 - sqrt(n) and n/2 + sqrt(n).

Clearly this term can also be described as a sum of binomial coefficients, but I don't know how to simplify it. I am less interested in the exact outcome (Although it would be great if I could get it), and more interested in the asymptotical lower bound. (Is this about Omega(2^n)? Omega(2^sqrt(n))? Something else?)