Let $A\_k$ be the $n\times n$ matrix defined by
$$
A\_k=\left[
\begin{array}{}
 1 & x\_1 & x\_1^2 & \cdots & x\_1^{n-2} & x\_1^k \\
 1 & x\_2 & x\_2^2 & \cdots & x\_2^{n-2} & x\_2^k \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
 1 & x\_n & x\_n^2 & \cdots & x\_n^{n-2} & x\_n^k
\end{array}\right].
$$
If $k=n-1$, the matrix $A\_{n-1}$ is called the Vandermonde matrix and the
 determinant of $A\_{n-1}$ is given by
$$
\det (A\_{n-1})= \prod\_{1 \mathop \le i \mathop < j \mathop \le n} \left({x\_j - x\_i}\right).
$$


My question is how to prove 
$$
\det (A\_n)= (x\_1+x\_2+\cdots + x\_n)\prod\_{1\leq i < j \leq n} \left({x\_j - x\_i}\right).\tag{1}
$$
 I know that $\det (A\_n)$ is a kind of [Schur polynomial](https://en.wikipedia.org/wiki/Schur_polynomial), but I don't know how to apply the Schur polynomial to prove Eq. $(1)$. 


Thanks in advance for your assistance.



Prove a special form of Schur polynomial identity
This is exactly Exercise 6.16 in [Darij Grinberg, *Notes on the combinatorial fundamentals of algebra*, 10 January 2019](https://github.com/darijgr/detnotes/releases/tag/2019-01-10), except that the order of columns in my exercise is the other way round (which is why I end up with $\prod\limits\_{1\leq i<j\leq n} \left(x\_i-x\_j\right)$ instead of $\prod\limits\_{1\leq i<j\leq n} \left(x\_j-x\_i\right)$). Of course, the difference is insubstantial: Reversing the order of the columns of an $n\times n$-matrix multiplies its determinant by $\left(-1\right)^{n\left(n-1\right)/2}$.


This is also (again, after the same reversal of the order of the columns) a consequence of "Corollary (The Bi-Alternant Formula)" in [John R. Stembridge, *A Concise Proof of the Littlewood-Richardson Rule*](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1n5), applied to $\mu = \left(1,0,0,\ldots,0\right)$ (with $n-1$ zeroes), since what he calls $a\_\rho$ is the usual Vandermonde determinant $\prod\limits\_{1\leq i<j\leq n} \left(x\_i-x\_j\right)$. This is the argument that Richard Stanley has been alluding to.


**EDIT:** The very same question has been posted, and answered by me, at <https://math.stackexchange.com/questions/3024496/computing-an-almost-vandermonde-matrix>.


