I am trying to understand the following pattern from the expansion of
$$(x - a_n)(x - a_{n - 1})\cdots(x - a_0)$$
Where $a_n\cdots a_0$ are distinct coefficients, I want to find the sum of the coefficients that are of terms with $x$ of degree $1$ and how a separate sequence can be formulated to find them.
I have tried applying the binomial theorem, as well as Vieta's formula to find a pattern that is similar to this one, but to no avail as it immediately gives the sum of the coefficients rather than showing the pattern that defines them.
Using $d_{n - 1} \cdots d_0$ which are the coefficients of $x$ with degree $n \geq 2$, I would like to define the pattern where
$$\frac{(x - a_n)(x - a_{n - 1}) \cdots (x - a_0) - (x^n + d_{n-1}x^{n - 1}\cdots + d_0x^2) + (a_n \cdots a_k)(-1)^{k - 1}}{x}$$
For example, Using $S_n$ which represents a sequence for the sum of our coefficient's, letting $n = 3$, would give
$$S_3 = a_3a_0 + a_{2}a_0 + a_3a_{2}$$
This is found by using the non - sequence formula I mentioned above.
Thus, define a formula for $S_n$