hi I have to write the following in closed form,
$$a_1 + a_2 + a_3 + a_1 v_{2}v_{1}+ a_1v_{3}v_{1}+ a_2v_{3}v_{2}+a_2v_{1}v_{2}+ a_3v_{1}v_{3}+a_3v_{2}v_{3}$$
$$ \sum_{(i,j)\in \mathcal{S}} a_i(1+ v_{j}v_{i})$$ where $$\mathcal{S}= \{ (i,j) \in \mathbb{Z}^2: i \neq j, 1\leq i,j \leq n=3 \}$$ in this case.
But I am not sure if it's correct, I think its not as it will give $$2 a_1 + 2a_2 + 2a_3 + a_1 v_{2}v_{1}+ a_1v_{3}v_{1}+ a_2v_{3}v_{2}+a_2v_{1}v_{2}+ a_3v_{1}v_{3}+a_3v_{2}v_{3}$$
Does anyone know how to fix this problem? or maybe its correct. Thanks
How about this?$$ \sum_{(i,j)\in \mathcal{S}} a_i\left(\frac12+ v_{j}v_{i}\right)$$