Suppose that we have two sequences of $n$ naturals, which we'll call $a$ and $b$. We can denote the $k$th value in the sequence by the term $a_k$ and $b_k$ respectively. We can then define a like multiplication for some number $v$. To do this, we pick $j$ terms from $a$, and at the same time take the terms with the same indexes from $b$. For example, if we pick 3 terms from $a$, and they are $a_3$, $a_5$, and $a_9$, then we would also pick $b_3$, $b_5$ and $b_9$. The like multiplication is then $$(v+a_3+a_5+a_9) \cdot (v+b_3+b_5+b_9)$$
In other words, the like multiplication for $v$ is just $v$ plus some terms from $a$ multiplied by $v$ plus the same terms from $b$. I would like to know the fastest way to find the sum of all possible like multiplications for some $v$ and two equal length sequences $a$ and $b$.
This is somewhat related to this question that I asked before. In particular, we may be able to eliminate $v$ from the equations using that question's answer.