In my project I needed an upper bound of binomial coefficient in terms of the gaussian density, so I derived the inequality
$$ \frac{1}{2^n} \binom{n}{\frac{n}{2}+\frac{\sqrt{n}}{2}t} \leq \sqrt\frac{2}{n\pi} e^{-(\frac{1}{2}-\frac{1}{n})t^2} $$
which holds for any $n\geq 1$ and $|t| \leq \sqrt{n}$.
Note that this bound is asymptotically tight at least in the regime $t=o(n^{1/4})$, which can be proved using the Stirling's approximation. But I needed an explicit and uniform upper bound that always works, so this is what I came up with.
I am pretty sure this kind of upper bound has already been studied in the literature, although a quick googling did not reveal anything. I would greatly appreciate for any help!