Consider a Hankel matrix $H \in \mathbb{R}^{n, n}$ with the following elements
$$h_{ij} = -\frac{4}{i+j} \quad \text{if } \quad i+j = \text{even}\\ h_{ij} = \frac{2}{i+j+1} + \frac{2}{i+j-1} \quad \text{if } \quad i+j = \text{odd}$$
Can we show that this matrix is PSD?
I implemented this matrix for different $n$ and computed the eigenvalues. Up to $n=12$, all eigenvalues are greater than zero. However, starting from $n=13$, some very small eigenvalues start to appear (of order $10^{-17}$). I suspect that this might be due to numerical stability since this also happens when computing the eigenvalues of the Hilbert matrix which we know is positive definite.
Edit: Thanks to the comment below, I realized that I made a mistake in the code and the matrix is not PSD (easily seen for $n=1$).