The history of the Vandermonde notation is described, in the context of the Vandermonde determinant, in section 2.1 of A case of mathematical eponymy: the Vandermonde determinant (2010). 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.
The following quotation from his 1772 paper shows how Vandermonde used this positional notation for coefficients to construct determinants:
Here is the original in French, from Mémoire sur l'Élimination, M. Vandermonde, Histoire de L'Academie Royale des Sciences, 1772, part two, pages 516-532.
