Rational approximation of $x^n$

Let $E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in
approximation of $x^n$ on $[\kern .3pt 0,1]$ by rational functions of type
$(k,k)$ with $k<n$. We show that in an appropriate limit $E_{kk}^{(n)} \sim
2\kern .3pt H^{k+1/2}$ independently of $n$, where $H \approx 1/9.28903$ is
Halphen's constant. This is the same formula as for minimax approximation of
$e^x$ on $(-\infty,0\kern .3pt]$.