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?
Yes, this is a special case of the de Moivre–Laplace theorem.
If it was $$A_n=\frac 1{2^n} \binom{n}{\frac{n}{2}+c \sqrt{n}}=\frac 1{2^n} \frac {\Gamma(n+1)}{\Gamma(\frac{n}{2}+c \sqrt{n}+1)\,\,\Gamma(\frac{n}{2}-c \sqrt{n}+1)}$$ taking logarithms, using three times Stirling approximation and exponentiating again $$A_n=\sqrt{\frac{2}{\pi n}}\,\, e^{-2 c^2}\,\left( 1-\frac{16 c^4-24 c^2+3}{12 n}+O\left(\frac{1}{n^2}\right)\right)$$ You could do the same kind of expansion using the bounds of the floor function.
