Binomial Theorem

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