Let \( \alpha = \displaystyle \sum_{r=0}^{n} (4r^{2} + 2r + 1)\, { }^{n}C_{r} \), and \( \beta = \left( \displaystyle \sum_{r=0}^{n} \frac{{ }^{n}C_{r}}{r+1} \right) + \frac{1}{n+1} \). If \( 140 < \frac{2\alpha}{\beta} < 281 \), then the value of \( n \) is _______.
