Vieta's formulae - looking for a plain English explanation of the general-case formulae I am working with a set of equations within which arise functions of the coefficients of a polynomial expressed in terms of the roots. The expressions get rather complicated as the order of the polynomial goes above 3, and so I want to write these coefficients using Vieta's formulae:
\begin{equation}
\sum_{1\leq i_1 < i_2 \cdots i_k \leq n}\left(\prod_{j=1}^k \alpha_{i_j}\right)
\end{equation}
However, I am struggling to disentangle this notation, in particular the expression beneath the summation and the notation subscripts for $\alpha$. I know what the result should be for a polynomial of order 1, 2, 3 4 and so on, but I would appreciate a plain-language description of how the notation describes this - and any suggestions as to how the expression could be made more straightforward to interpret?

This particular formula is for a particular $k$.
The summation is for every strictly increasing sequence of $k$ elements, in which each element ranges from $1$ to $n$ inclusive. The inequality $1\le i_1 < i_2 < \ldots < i_k \le n$ denotes that the sequence $(i_1, i_2, \ldots, i_k)$ should have $k$ elements, the elements are strictly increasing, and the elements are between $1$ and $n$ inclusive.
There are $\binom n k$ such sequences, so there are $\binom nk$ summands. Each summand is a product, which may be written as
$$\prod_{j=1}^k \alpha_{i_j} = \prod_{i\in\{i_1, i_2,\ldots,i_k\}}\alpha_i.$$
The product means that for each unique sequence, the sequence would choose $k$ of the $n$ roots, and take their product.
For example, $k = 4$ and $n=6$, one of the sequences is $(1,2,3,5)$, and the sequence corresponds to the product $\prod_{i\in\{1,2,3,5\}} \alpha_i = \alpha_1\alpha_2\alpha_3\alpha_5$. There are $\binom64 = 15$ such products to sum.
So I may reword the summation and product sign as
$$\sum_{S : k-\text{sequence which is strictly increasing between }1\text{ and }n} \left(\prod_{i\in S}\alpha_i\right)$$
