  We describe a method to accelerate the numerical computation of the
coefficients of the polynomials $P_k(x)$ that appear in the conjectured
asymptotics of the $2k$-th moment of the Riemann zeta function. We carried out
our method to compute the moment polynomials for $k \leq 13$, and used these to
experimentally test conjectures for the moments up to height $10^8$.
