Compute the determinant (similar to Vandermonde)

Question

Compute the determinant of the matrix follows, $$\begin{pmatrix}1&2&\cdots&n\\1^2&2^2&\cdots&n^2\\\vdots&\vdots&\ddots&\vdots\\1^n&2^n&\cdots&n^n\end{pmatrix}$$

Thanks in advance.

Answer 1

It is Vandermonde for $x_i=i$ can be also written as $$\prod_{k=1}^{n+2} (k-1)! $$ there is a special function for that called Barnes G-function. In terms of it $$ G(n+2) $$