Double exchange interraction

matematikuvol

In double exchange interraction spin [tex]S_{uk}=S\pm \frac{1}{2}[/tex]. Isn't there 2 jumping electrons? Why hamiltonian is written in way
[tex]\hat{H}\approx \sum_{n=0}^{2S}J_{n}(S)(\vec{S}_i\cdot\vec{S}_j)^{n}[/tex]
Why from 0 to 2S?