\displaystyle $\begin{align*}
D_n & = \left| \begin{array}{rrrrrrr}
1 & 1 & 1 & \ldots & 1 \\
2 & 3 & 4 & \ldots & n+2 \\
 2^2 & 3^2 & 4^2 & \ldots & (n+2)^2 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
2^n & 3^n & 4^n & \ldots & (n+2)^n \\
\end{array} \right|\end{align*}