Binomial Theorem

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