I've been looking at a certain type of sum, the first case being $\sum_{i<j} {a_i a_j}$, and trying to simplify it/generalise it. It would be far more useful to write them more explicitly, in terms of powers of sums of terms, etc. For the first case, I've found, that by considering $\left( \sum_i a_i \right)^2$, and writing that sum out in full;
$$ \begin{aligned} & \ \ \ \ \ a_1(a_1+a_2+a_3 \cdots) \\ &+ a_2(a_1+a_2+a_3 \cdots) \\ &+ a_3(a_1+a_2+a_3 \cdots) \end{aligned} $$
the diagonal can be written as $\sum_i{a_i^2}$, while the 'triangles' left behind are equal to $2\sum_{i<j} {a_i a_j}$.
My question is how would I generalise this, for any amount of variables? For example, I have tried the same method for $\sum_{i<j<k}{a_i a_j a_k}$ to no avail. Has this been studied anywhere else? Thanks in advance.
What you are looking at are relations between degree $3$ (more generally, degree $n$) symmetric polynomials. For the case of 3 variables, first define the following polynomials for $k\geq1$. \begin{align*} S_k&=\sum_{i}a_i^k\\ T_k&=\sum_{i_1,\ldots,i_k\text{ distinct}}a_{i_1}\ldots a_{i_k} \end{align*} Now we have \begin{align*} S_1^3 &= S_1 S_1^2\\ &= S_1(S_2+T_2)\text{ (using your computation)}\\ &= S_1S_2+S_1T_2\\ &= S_1S_2+\sum_{i\neq j}a_i^2a_j+\sum_{i\neq j}a_ia_j^2+T_3\\ &=S_1S_2+T_3+2\sum_{i\neq j}a_ia_j^2\\ &=S_1S_2+T_3+2(S_1S_2-S_3)\\ &=3S_1S_2-2S_3+T_3 \end{align*} Solving for $T_3$ (which is $6$ times the sum you're after), we obtain $$T_3=S_1^3-3S_1S_2+2S_3$$
Edit: I found out about Newton's identities after digging through the Wikipedia page of symmetric polynomials, and those are precisely about relations of the above type.