My maths book uses this kind of notation for sums (circled in red) but doesn't explain what it means. I don't know what it's called or where to find resources explaining it.
Could someone explain it to me or point out to somewhere where I can read more about it? Just naming it so I can search it would be helpful.
Screenshot of my maths book here: https://i.stack.imgur.com/dIFeO.png
The notation of
$$\sum_{S(k_1, k_2, \ldots)}$$
where $S$ is a statement or equality in terms of $k_1, k_2, \ldots$ means "sum over the values that make the statement true". So in this case, it's "sum over values of $k_1$ to $k_n$ that add up to $k$".
There is an implicit restriction on the values here that they are positive (or at least non-negative) integers, although this may not always be true - for example,
$$\sum_{i \in U} a_i$$
is a sum of $a_i$ over all elements $i$ in the set $U$, but it doesn't even state what kind of set $U$ is.
It's also implied that the free variables - the ones we're going to vary in our sum - are the $k_i$ - and that the $k$ is a fixed parameter (this should be clear from the context since $k$ appears in parts of the expression outside of the sigma). As noted in the comments, the more general way to express this might be something like:
$$\sum_{S(k_1, k_2, \ldots ; \theta_1, \theta_2, \ldots)}$$
I'd also note that you could expand the expression out into a more explicit representation, but this can make things more, rather than less, complicated. In the case of your particular sum, for example (and replacing the expression inside the sum with a generic term):
$$\sum_{k_1 + k_2 + \ldots + k_n = k} a(k_1, k_2, \ldots, k_n) = \sum_{k_1 = 0}^k \sum_{k_2 = 0}^{k - k_1} \ldots \sum_{k_n = 0}^{k - k_1 - k_2 - \ldots - k_{n-1}} a(k_1, k_2, \ldots, k_n)$$
Isn't that horrendous?
