$$\sum _{i=0}^{\infty }\sin \left(ix\right)\prod _{j=0}^N\left(j^2+j\right)$$ $$\sum _{i=0}^{\infty }\left(\frac{\sin \left(\raisebox{1ex}{$i\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{i!}\right)\prod _{j=0}^N\left(\frac{{\sigma }_j+\phi }{{\left(x+y\right)}^j}+j\right)$$

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