Let α=Σ_k=0^n(((^n C_k)^2)/(k+1)) and β=Σ_k=0^n-1((^n C_k^n C_k+1)/(k+2)). If 5 α=6 β, then n equals ____.

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