  In order to better understand the structure of classical rings of invariants
for binary forms, Dixmier proposed, as a conjectural homogeneous system of
parameters, an explicit collection of invariants previously studied by Hilbert.
We generalize Dixmier's collection and show that a particular subfamily is
algebraically independent. Our proof relies on showing certain alternating sums
of products of binomial coefficients are nonzero. Along the way we provide a
very elementary proof \`a la Racah, namely, only using the Chu-Vandermonde
Theorem, for Dixon's Summation Theorem. We also provide explicit computations
of invariants, for the binary octavic, which can serve as ideal introductory
examples to Gordan's 1868 method in classical invariant theory.
