Determinant of a matrix with integer power entries Define $$D(n,k)=\begin{vmatrix} 1^k & 2^k&\cdots& n^k\\2^k&3^k&\cdots &(n+1)^k\\ \vdots&\vdots&\ddots&\vdots\\ n^k&(n+1)^k&\cdots&(2n-1)^k 
\end{vmatrix}.$$
I'm asking:


*

*Calculate $D(1,1), D(2,1), D(3,1), D(4,1)$.

*Show that $D(n,2)=0$ for $n>3$.

*Show that $D(n,k)=0$ for $n>k+1$


First one is not difficult. I thought that maybe this calculation could help me to attack the second and third question. Is not the case. For $n=4$ one can verify that this determinant is truly zero by making zeros on the first row an then calculating a $3\times 3$ determinant. However, it doesn't help for found a generalized method for demonstrate the assertion. It's pretty clear (I think) that induction is the ideal method but I'm not sure on how to proceed. I try to use the fact that the matrix is symmetric, but again without success.
Thanks in advance! 
 A: For point (2), if you can find quantities $a, b, c, d$, not all zero, such that
$$ a m^2 + b(m+1)^2 + c(m+2)^2 + d(m+3)^2 = 0$$ for any integer $1 \leq m \leq n$, then this gives you a nontrivial linear combination of the first four rows in $D(n, 2)$ that equals zero, which proves $\det D(n, 2) = 0$. The above equation gives a system of three equations:
$$ \begin{align*}
a + b + c + d &= 0 \tag{matching coefficients of $m^2$} \\
2b + 4c + 6d &= 0 \tag{matching coefficients of $m$} \\
b + 4c + 9d &= 0 \tag{matching constants}
\end{align*} $$
which must have a nonzero solution, as it's an underspecified homogeneous linear system (fewer equations than variables). The solutions are when $(a, b, c, d)$ is a scalar multiple of $(1, -3, 3, -1)$, but the exact solution doesn't matter; only the existence of a solution does.
Point (3) is similar. You can show the existence of $k 
+ 2$ quantities $a_0, \ldots, a_{k+1}$ such that $a_0 m^k + a_1 (m+1)^k + \cdots + a_{k+1} (m+k+1)^k = 0.$ Matching coefficients for powers of $m$ gives an underspecified linear system of $k+1$ equations, which must have a nonzero solution. This gives a nontrivial zero linear combination of the first $k+2$ rows in $D(n, k)$.
