An $n$-variable Boolean function $f:\{0,1\}^{n} \rightarrow\{0,1\}$ is called symmetric if its value depends only on the number of $1 \text{'s}$ in the input. Let $\sigma_{n}$ denote the number of such functions.
- Calculate the value of $\sigma_{4}$.
- Derive an expression for $\sigma_{n}$ in terms of $n$.