Let the coefficient of x^r in the expansion of (x+3)^n-1+(x+3)^n-2(x+2)+(x+3)^n-3(x+2)^2+… … …+(x+2)^n-1 be α_r. If Σ_r=0^n α_r=β^n-γ^n, β, γ ∈ N,…

Let the coefficient of $\displaystyle \mathrm{x}^{\mathrm{r}}$ in the expansion of $\displaystyle (x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots+(x+2)^{n-1}$ be $\displaystyle \alpha_r$. If $\displaystyle \sum_{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}$, then the value of $\displaystyle \beta^2+\gamma^2$ equals $\displaystyle \_\_\_\_$.