The history of the Vandermonde notation is described, in the context of the Vandermonde determinant, in A case of mathematical eponymy: the Vandermonde determinant, section 2.1. It seems Lebesgue didn't like it because it could have induced a mix-up between indices and exponents, and that may be a reason it did not survive. Leibniz used a similar notation.
http://ilorentz.org/beenakker/MO/Vandermonde.png
The following quotation from his 1771 paper shows how Vandermonde used this positional notation for coefficients to construct determinants:
http://ilorentz.org/beenakker/MO/Vandermonde_2.png