  We prove a conjecture of Okada giving an exact formula for a certain
statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{\lambda \vdash n}
f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}(h_u^2 - i^2) =
\frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{r+1} \prod_{j=0}^{r} (n-j), where
$f_{\lambda}$ is the number of standard Young tableaux of shape $\lambda$ and
$h_u$ is the hook length of the square $u$ of the Young diagram of $\lambda$.
We also obtain other similar formulas.
