As observed in comments, the problem is equivalent to finding the inverse of the matrices $$H_\lambda:=\Big[{1\over i+j+\lambda}\Big]_{1\le i\le m\atop 1\le j\le m},$$ 
for $\lambda=-1/2$ and $\lambda=-3/2$ and order $m=\lceil n/2\rceil$, respectively $m=\lfloor n/2\rfloor$. These are also particular cases of Cauchy matrices, whose inverses admit the explicit  Schetcher's formula  quoted in the linked question (see the wiki article on [Cauchy matrices][1] and the various answers in the linked question). As a matter of  fact, in this particular case the inversion formula takes quite a simpler form, and it turns out that $H_\lambda$ entries are products of linear factors $\lambda+k$, with $2\le k\le 2m$ and multiplicity not larger than $2$. Precisely

$$H^{-1}_\lambda:={1\over(n-1)!^2}\bigg[(-1)^{i+j}{m-1\choose i-1}{m-1\choose j-1}(\lambda+i+1)^{\overline m}(\lambda+j+1)^{\overline m}(\lambda+i+j)^{-1}\bigg]_{1\le i\le m\atop 1\le j\le m}$$ 
where $\overline m$ denote a rising factorial.




[1]:https://en.wikipedia.org/wiki/Cauchy_matrix#Cauchy_determinants