Let's say I know the $N$ roots $\boldsymbol r$ of a polynomial $p_N(x)$ and I want to compute the coefficients $\boldsymbol \alpha $ of the representation in monomials, i.e., $$p_N(x) = \sum_{j=0}^N \alpha_j x^j. $$
Vieta's formulas give an readily understood way of how to do this, but I struggle with the (efficient) implementation. Throughout the computation of the $\alpha_j$, the roots are combined in $2^N -1$ different ways. To see this, consider the power set of $\{1, \dots, N \}$ which has $2^N$ elements. The products $$ \prod_{i \in S \subset\{1, \dots, N \}} r_i$$ are then summed up for subsets $S$ of same cardinality $n>0$ to give the $\alpha_{N-n}$ coefficient.
I was wondering whether someone sees an alternative way of computing the sums of products than enumerating the whole powerset (which consumes drastic amounts of memory just for storing them for $N \sim \mathcal{O}(50)$) and then summing the products of the subsets with same cardinality.
