\begin{equation} A = \frac{(\frac12-\frac{1}{n})(\frac12-\frac{2}{n})...(\frac12-\frac{t-1}{n})}{(\frac{1}{2}+\frac{1}{n})(\frac{1}{2}+\frac{2}{n})... (\frac{1}{2}+\frac{t}{n})} \end{equation}
Can we get an asymptotic estimate for above A? Assume $n \gg 1$ and $t \ll n$