Consider following partial Vandermonde type, circulant matrix $\begin{bmatrix} x_1 & x_2 & 0 & \dots & 0 & x_n\\ x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\ \vdots &\vdots &\vdots &\ddots &\vdots &\vdots\\ x_1^n & 0 & 0 & \dots & x_{n-1}^n & x_n^n\\ \end{bmatrix}$.

Is there a closed form expression for this determinant like Vandermonde determinant expression?