Coefficients of the polynomial approximation of $|x|$

Question

I am looking for a polynomial $p_n$ of degree at most $n$ that will approximate $|x|$ in the interval $[-1,1]$ with $\||x|-p_n\|_\infty=O(\frac{1}{n})$ and will have coefficients whose absolute value is monotonically decreasing with $n$ as fast as possible. Something of the sort $|a_i|=O(1/i!)$, where $a_0,\cdots,a_n$ are the coefficients of $p_n$. Is there some known result of this spirit?