$$ \det\begin{bmatrix} s_0&s_1&s_2&\cdots&s_{n-1}\\ s_1&s_2&s_3&\cdots&s_n\\ s_2&s_3&s_4&\cdots&s_{n+1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ s_{n-1}&s_n&s_{n+1}&\cdots&s_{2n-2}\\ \end{bmatrix}$$
where $$s_k = x_1^k + x_2^k + \dots + x_n^k $$
I solved the task. I used Cauchy's theorem on the determinant of the product. Let the above matrix be A.
$\det(A) = \det(BC) = \det(B) \det (C)$, where C is Vandermonde matrix and B it's a transposition. So, the determinant is Vandermonde determinant to the power of 2. Can someone check my reasoning?