Notation of sigmas When one writes $$\sum_{i+j=4} a_i a_j$$ is that then equal to $$a_0a_4 + a_1a_3 + a_2^2$$ or $$\sum_{i=0}^4 a_i a_{4-i}=2a_0a_4+2a_1a_3 + a_2^2$$ I suddenly got confused while writing these sigmas down. Perhaps it depends on context or perhaps not? Thanks for any help. 
 A: I think that there is no ambiguity on the sum. When you write
$$ \sum_{i+j = 4} a_i a_j$$
you are summing over the set of all solutions of $i+j = 4$. The solutions are


*

*$i = 0$ and $j = 4$

*$i = 1$ and $j = 3$

*$i = 2$ and $j = 2$

*$i = 3$ and $j = 1$

*$i = 4$ and $j = 0$


So the sum should result:
$$ \sum_{i+j = 4} a_i a_j = a_0a_4 + a_1a_3 + a_2a_2 + a_3a_1 + a_4a_0 = 2a_0a_4 + 2a_1a_3 + a_2^2$$
The confusions only happens because you are indexing $a$ twice, how would you interpret the following sum?
$$ \sum_{i+j = 4} a_i b_j$$

Most of the places where sums like this appears on the indexing set of the summation, it on a product of generating functions / power series, like this:
$$ \left ( \sum_{i=0}^\infty a_i x^i  \right ) \left ( \sum_{j=0}^\infty b_j x^j \right ) = \sum_{n=0}^\infty \left ( \sum_{i+j = n} a_i b_j \right )x^n $$
Note that the coefficient of $x^n$ on the right-hand side is a sum like yours. The correct interpretation on this case should be:
$$ \sum_{i+j = n} a_i b_j = a_0 b_n + a_1 b_{n-1} + a_2 b_{n-2} + \dotsc + a_n b_0 $$
This also expands nicely for multiple variables, so we have things like the multinomial theorem
$$\left ( x_1 + x_2 + \dotsc + x_m \right )^n = \sum_{k_1 + k_2 + \dotsc + k_m = n} \binom{n}{k_1,k_2,\dotsc,k_m} \prod_{t=1}^{m}x_t^{k_t}$$
here the sum have to be interpreted as being over all the $m$-tuples $(k_1, k_2, \dotsc, k_m)$ satisfying the equation. So this is the standard interpretation for sums of this type.

Of course that maybe you want to sum over the set that you have presented, but then I think is best to indicate explicitly, as that's not the common case. One way to do it would be:
$$\sum_{\substack{i+j = 4 \\ i \leq j}} a_i a_j = a_0a_4 + a_1 a_3 + a_2^2$$
Note that this can also be easily generalized.
