Exact evaluation of <Tr S^p> is here performed for real symmetric matrices S of arbitrary order n, up to some integer p, where the matrix entries are independent identically distributed random variables, with an arbitrary probability distribution. These expectations are polynomials in the moments of the matrix entries; they provide useful information on the spectral density of the ensemble in the large n limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit.
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