Binomial coefficient asymptotics

Question

What is the probability that the number of heads in $n$ fair coin tosses is exactly $\lfloor n/2 + c\sqrt{n} \rfloor$ for $c \leq O(1)$, $n > \omega(1)$?

Of course the answer is $$ \frac{1}{2^n} \binom{n}{\lfloor n/2+c\sqrt{n} \rfloor}.$$

I have estimated this and found it to be

$$\frac{1}{\sqrt{n}} e^{-\Theta(c^2)}.$$

Is this correct? Is there a good name/reference for this?

Answer 1

Yes, this is a special case of the de Moivre–Laplace theorem.