On Louchard's Asymptotic Series

Recently G. Louchard obtained an asymptotic series ${\sum }_{j=0}^{\mathrm{\infty }}\frac{I_j}{n^j}$ for the integral ${\int }_0^1[x^n+(1-x{)}^n{]}^{\frac{1}{n}}dx$ as $n\to \mathrm{\infty }$, and computed $I_j$ for $j\le 5$ in terms of values of the Riemann zeta function. An interesting feature of the computation is that the $I_j$ are first obtained in terms of alternating multiple zeta values, but then everything except products of ordinary zeta values cancels out. We obtain similar formulas for $I_n$, $6\le n\le 9$,andconjectureageneralformulafor$I_n$ in terms of alternating multiple zeta values. We also conjecture that $I_n$ is a rational polynomial in the ordinary zeta values.