In the last MO question , the following matrix is given: $$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$ and its inverse has been discussed. Now the problem is further extended to a more general form: $$M'_{ij}=\frac{(z+1)^{i+j-1}-(z -1)^{i+j-1}}{i+j-1}$$ where $-1 \le z \le 1$.

Similarly, is there an explicit formula for the inverse of $M'$? How about if we remove the limitation on $z$ and assume $-\infty \lt z \lt \infty$?