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Good evening,

It is known that the determinants $D_{k}=\begin{vmatrix}(i+j)^{k}\end{vmatrix}_{1 \leq i,j \leq n}$ are zero for $k<n-1$ and $D_{n-1}=(-1)^{n(n-1)/2}((n-1)!)^{n}$: Newton's binomial formula shows that the former matrices have rank no more than $k+1$, while it expresses the latter as the product of two matrices whose determinant is nearly Vandermonde.

Is there an easy way to compute $D_{k}$ for $k>n$? This method doesn't seem to work any more.

Thanks

Tig la Pomme
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    Usually I advise OEIS for questions like this, but this particular sequence doesn't show up prominently. One case I did find was for $k=n$, which appears as sequence A092415. (The first entry is the trivial case $n=k=0.$) It does not, however, indicate a closed-form and so it's entirely possible that one is not known. – Semiclassical Apr 30 '17 at 21:01

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