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I understand normal sigma notation but what does it mean when we place under a sum the restriction that $i + j + k = n$, for example? Is this simply $3$ sums in disguise or is it something else?

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It is basically what you said:

the restriction that i + j + k = n

So if your sum looks like

$$ \sum_{i+j+k=n} g(i,j,k), $$

you sum over all $g(i,j,k)$ where $(i,j,k)$ satisfy the condition. That means you need to find the set of all combinations $\{(i,j,k) | i + j + k = n\}$.

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    $\begingroup$ Both the summation and your set notation are missing a specification of what $i,j,k$ are ranging over. Since the set needs to be finite for the sum to be defined without further (convergence) conditions, it may be assumed that each was meant to independently range over $\Bbb N$, but this needs to be made explicit to ensure these expressions are unambiguous. $\endgroup$ – Marc van Leeuwen Oct 3 '13 at 14:23
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    $\begingroup$ You are right, of course. But I don't know these conditions, they would normally come become clear for a specific problem. I only put in my answer the information @Stopwatch gave in the question without making any general assumptions about the range of $i,j,k$ or about convergence. $\endgroup$ – Lisa Oct 3 '13 at 14:32
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It means sum over all possible triples $(i,j,k)$ such that $i+j+k=n$. Presumably, to ensure finiteness, we are only considering nonnegative integers $i,j,$ and $k$. For example

$$\sum_{i+j+k=2}f(i,j,k)=f(0,0,2)+f(0,2,0)+f(2,0,0)+f(0,1,1)+f(1,0,1)+f(1,1,0)$$

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