Let $a_n = \displaystyle\sum_{k=0}^{n-1}\binom{2n-1}{k}$ and $b_n=\displaystyle\sum_{k=n}^{2n-1}\binom{2n-1}{k}$. Then, $$a_n + b_n = \displaystyle\sum_{k=0}^{2n-1}\binom{2n-1}{k}=2^{2n-1}$$

Meanwhile, because $\binom{2n-1}{k}=\binom{2n-1}{2n-1-k}$, it is possible to pair every term in $a_n$ with one from $b_n$ of equal value, and vice versa. This means $a_n=b_n$.

Therefore, the final answer is $\boxed{2^{2n-2}}$.