Let \( \alpha = \displaystyle\sum_{k=0}^{n} \left( \dfrac{({}^{n}C_{k})}{k+1} \right)^{2} \) and \( \beta =\displaystyle \sum_{k=0}^{n-1} \left( \dfrac{{}^{n}C_{k}\,{}^{n}C_{k+1}}{k+2} \right) \). If \( 5\alpha = 6\beta \), then \( n \) equals _______.
