TITLE: Find the inverse of a matrix that is very similar to the Hilbert matrix
QUESTION [10 upvotes]: The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely, 
$$M_{ij}=\frac{1+(-1)^{i+j}}{i+j-1}.$$
Does anyone know the explicit formula for the inverse of M? Please provide some relevant reference if available.
A more general form for this question can be found here.

REPLY [3 votes]: These matrices have been considered here.