Let $n > 0$, then :
$$ \det \left( {2n \choose n+i-j} \right)_{i,j=0}^{n-1} = \prod_{i=0}^{n-1} \frac{2n+i \choose n}{n+i \choose n} $$
The LHS appears to be the determinant of a symetric Toeplitz matrix with binomial coefficients.
Does anyone have an idea on how to proove this formula ?