Eigenvalues of Hadamard powers of large symmetric Pascal matrices

Let Sn be the positive real symmetric matrix of order n with (i, j) entry equal to i+j-2j-1, and let x be a positive real number. Eigenvalues of the Hadamard (or entry wise) power Sn(x) are considered. In particular for k a positive integer, it is shown that both the Perron root and the trace of Sn(k) are approximately equal to 4k4k-12n-2n-1k.