Fourier coefficients of $f\left(z\right)=\frac{1}{1+\cos z}$ through Laurent series

Question

I had to find the Fourier coefficients of this simply periodic function $$f\left(z\right)=\frac{1}{1+\cos (z)},$$ I proceeded considering the $w=exp(iz)$ and considering the Laurent expansion of the function $F(w)$ $$F\left(w\right)=\frac{1}{1+\frac{1}{2w}\left(w^{2}+1\right)}=\frac{2w}{\left(w+1\right)^{2}},$$ If I'm not making a mistake, this function has a Laurent expansion in zero inside the unit circle given by $$F\left(w\right)=\sum\left(-1\right)^{n}2\left(n+1\right)w^{n+1},\,\,\,n\geq0$$ So writing back fourier expansion for $f(z)$ I have the following Fourier expansion $$f\left(z\right)=\sum c_{n}e^{inz},\,\,\,\,\,c_{n}=\left(-1\right)^{n}2n,\,\,\,\,n\geq1$$ I think everything is fine but when I'm doing the same analysis with an equivalent formulation of $f(z)$ I have a different result. Can anybody check for me if those I found are the real Fourier coefficients of my $f(z)$?