\frac{1}{\cos \pi z}=\frac{4}{\pi}\sum_{k=0}^{\infty} (-1)^k \frac{2k+1}{(2k+1)^2-(2z)^2}=\frac{4}{\pi}\sum_{k=1}^{\infty} (-1)^{k-1} \frac{2k-1}{(2k-1)^2-(2z)^2}