While going through an exercise, I ended up proving the result for the determinant of this 'Vandermonde-like' matrix for $n \geq 2, \ k \geq n-1$:
$$\det\begin{pmatrix}1& a_1 & \cdots & a_1^{n-1}& a_1^k\\ 1& a_2 & \cdots & a_2^{n-1}& a_2^k\\ \vdots &\vdots & & \vdots & \vdots\\ 1& a_n & \cdots & a_n^{n-1}& a_n^k \end{pmatrix}=\prod_{i > j}(x_{i}-x_{j})\sum_{\alpha_{1}+...+\alpha_{n} = k-n+1}(x_{1}^{\alpha_{1}}...x_{n}^{\alpha_{n}})$$
What I would like to know if it has any special name, or where I can find other cool variants of the Vandermonde matrix.