Bitcoin Core Development – PinSketch Why are sketches made up of sums of odd powers?

The following is from BIP-330 (aka Erlay):

A short identity sketch of capacity c consists of a sequence of c field elements. The first is the sum of all short IDs in the set, the second is the sum of all short IDs to the power of three, and the third is the sum of all short IDs to the power of five. The sum up to the last element is: (2c-1) Secondary forces.

why the first c weird Is force chosen instead of first c? How can we reconstruct the c elements from the first c odd power sum? It’s rather straightforward to use Newton’s identity if you have the first c, but it’s not clear to me how to do so with a strange c.