Let the coefficients of \( x^{-1} \) and \( x^{-3} \) in the expansion of \( \left(2x^{1/5}-\frac{1}{x^{1/5}}\right)^{15} \), \( x>0 \), be \( m \) and \( n \) respectively. If \( r \) is a positive integer such that \( mn^{2}=\,^{15}\!C_{r}\cdot 2^{r} \), then the value of \( r \) is equal to ______.
