Calculate the determinant of matrix $\{ a_{ij} = s^{i+j-2}\}$ where $s^k = a_1^k + a_2^k+\ldots +a_n^k$

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?

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