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The matrix is $n\times n$ matrix and $a_{ij} = s^{i+j-2}$ where $s^k = a_1^k + a_2^k+\ldots +a_n^k$.

The determination of is the sum of $a_{i1}^0a_{i2}^1\ldots a_{in}^{n-1}(-1)^{N(i_1i_2\ldots i_n)}V$, where $V$ is the Vandermonde determination and $i_1,i_2,\ldots, i_n$ is a permutation of $1,2,\ldots, n$.

Also, take $1,2,3,\ldots,n$ and $1,3,2,\ldots,n$ as an example, their sum is $(1-\frac{a_2}{a_3})a_1^0a_2^1\ldots a_n^{n-1}V$.

I don't know what to do now. Any hint?

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    $\begingroup$ Hint: Your matrix is the product of two versions of the Vandermonde matrix. Just write $a_{ij}$ as $a_1^{i-1} a_1^{j-1} + a_2^{i-1} a_2^{j-1} + \cdots + a_n^{i-1} a_n^{j-1}$. $\endgroup$ – darij grinberg Mar 10 at 3:47

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