  We give a new combinatorial proof of Parraud's formula for the asymptotic
expansion in powers of $1/N^2$ for the expected trace of polynomials of several
independent $N \times N$ GUE matrices, which expresses the result using a
mixture of free difference quotients, introducing new freely independent
semicircular variables, and integration with respect to parameters. Our
approach streamlines the statement of the formula while clarifying its
relationship to the combinatorial genus expansion.
