Sum of the products of the subsets

Consider the following product:

$\begin{aligned} \prod_{k=1}^n \left(x+a_k\right) & = x^n + x^{n-1} \sum_{k=1}^n a_k + x^{n-2} \sum_{j,k=1, j \ne k}^n a_ja_k + x^{n-3} \sum_{i,j,k=1, i \ne j \ne k}^n a_ia_ja_k + \cdots + \prod_{k=1}^n a_k \\ & = x^n + c_{n-1}x^{n-1} + c_{n-2}x^{n-2} + c_{n-3}x^{n-3} + \cdots + c_2 x^2 + c_1 x + c_0 \end{aligned}$

We note that $\displaystyle \sum_{k=0}^{n-1} c_k = S_a$ is the sum of products of each non-empty subset of the set $\{a_1, a_2, a_3, ... a_n\}$ . Replacing $x=1$ and $a_k = k$ , we have:

$\begin{aligned} \prod_{k=1}^n \left(1+k\right) & = 1 + S \\ \implies S+1 & = (n+1)! & \small \color{#3D99F6} \text{For }n = 2017 \\ S + 1 & = 2018! \\ \implies f(S+1) & = \boxed{2018} \end{aligned}$