Any non-empty subset of $\Psi = \{n+1, n+2, \dots, 2n\}$ is obviously a sum-free subset of $\Omega$, and there are $2^n - 1$ of these. Also, every non-empty subset of the set $\Phi$ of odd numbers in $\Omega$ is also a sum-free subset and there are $2^n - 1$ of these. The number of common subsets in the union of these two collections is $2^{\lceil n/2 \rceil} - 1$. So $\Omega$ has at least $2(2^n - 1) - (2^{\lceil n/2 \rceil} - 1) > 2^n$ sum-free subsets. Alternatively, for any $n \ge 2$, $\{1, 4\}$ and $\{2, 3\}$ are sum-free subsets of $\Omega$ and they are neither subsets of $\Psi$ nor of $\Phi$. Together with the $2^n - 1$ non-empty subsets of $\Psi$ or of $\Phi$ they provide the required $2^n + 1$ sum-free subsets.