Sills–Wang characteristic-polynomial conjecture for the Wiener-index matrix

Let $A_k=(a_{ij})$ be the $k\times k$ matrix with $a_{ij}=\frac{1}{2}|i-j|$, and let $I_k$ be the $k\times k$ identity matrix. Define $C_k(\lambda)=\det(A_k-\lambda I_k)$ to be the characteristic polynomial of $A_k$. Sills–Wang's characteristic-polynomial conjecture.

$$C_k(\lambda)=(-1)^k\lambda^k\left(1-\frac{k}{4}\sum_{j=1}^{k-1}\frac{j}{j+1}\binom{k+j}{2j+1}\lambda^{-j-1}\right).$$

The formula gives an explicit expression for the characteristic polynomial and was proposed to obtain sharper information about the largest eigenvalue of $A_k$. The paper proves this conjecture, so its database status is solved.