In my recent MO question, Darij Grinberg mentioned a closely related (structure-wise) determinant, that is, $$\det\left(x_{\min\{i,j\}}\right)_{i,j}^{1,m}=x_1(x_2-x_1)(x_3-x_2)\cdots(x_m-x_{m-1}).$$ In particular, $\det(\min\{i,j\})=1$. This prompted me to ask:

QUESTION.Is this true? The characteristic polynomial of the matrix $\mathbf{N}_m=(\min\{i,j\})_1^m$ equals $$P_m(\lambda)=\sum_{k=0}^m(-1)^{m-k}\binom{2m-k}k\,\lambda^k.$$

**Remark 1.** It's worth pointing out that
$$P_m(\lambda^2)=(-1)^m\lambda^{2m}\,U_{2m}\left(\frac1{2\lambda}\right),$$
where $U$ is *Chebyshev polynomial of the second kind.*

**Remark 2.** Needless to say, $P_m(\lambda)$ implies a formula for all elementary polynomials $\mathbf{e}_j$ of the roots of characteristic polynomial for the given matrix $\mathbf{N}_m$.

**Remark 3.** As a "fun" aside, the entries of the matrix $\mathbf{N}_m$ offer a systematic way of calculating the total number of squares (of all sizes) in an $m\times m$ grid; i.e.
$$\sum_{i,j=1}^m\min\{i,j\}=1^2+2^2+\cdots+m^2.$$