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