Summation Index Confusion I've seen various confusing variations of the summation signs, so can anyone give me clarification on them? I understand the most common ones from calculus class, for example:
$$\sum^\infty_{i=1}\frac{1}{i}\to\infty$$
The rest is very confusing, as all the indices and whatnot are on the bottom of the summation, sometimes with nothing on the top. For example, the multinomial theorem:
$$(x_1+\dots +x_k)^n=\sum_{a_1,a_2,\dots ,a_k\geq 0}
{n\choose{a_1,\dots ,a_k}}x_1^{a_1}x_2^{a_2}\cdots x_k^{a_k}$$
Or another formula to calculate $(1+y)^n$ when $n$ isn't necessarily a natural number:
$$(1+x)^n=\sum_{i\geq0}{n\choose{i}}x^i$$
There are also cases of the following exemplar types of summations:
$$\underset{i\text{ odd}}{\sum_{i=0}^n}\dots$$
$$\sum_{a_1=n}^n\dots$$
Can someone please explain what all these indications of summation indices mean? Thank you.

TZakrevskiy’s answer covers all of your examples except the one from the multinomial theorem. That one is abbreviated, and you simply have to know what’s intended or infer it from the context: the summation is taken over all $k$-tuples $\langle a_1,\ldots,a_k\rangle$ of non-negative integers satisfying the condition that $a_1+a_2+\ldots+a_k=n$. If the condition were written out in full, the summation would look like
$$\huge\sum_{{a_1,\ldots,a_k\in\Bbb N}\atop{a_1+\ldots+a_k=n}}\ldots\;.$$
(Note that my $\Bbb N$ includes $0$.)
Added example: Let $n=2$ and $k=3$. The ordered triples $\langle a_1,a_2,a_3\rangle$ that satisfy $a_1+a_2+a_3=2$ are:
$$\begin{array}{ccc}
a_1&a_2&a_3\\ \hline
0&0&2\\
0&2&0\\
2&0&0\\
0&1&1\\
1&0&1\\
1&1&0
\end{array}$$
Thus, the sum in question is
$$\begin{align*}
\binom2{0,0,2}x_1^0x_2^0x_3^2&+\binom2{0,2,0}x_1^0x_2^2x_3^0+\binom2{2,0,0}x_1^2x_2^0x_3^0\\
&+\binom2{0,1,1}x_1^0x_2^1x_3^1+\binom2{1,0,1}x_1^1x_2^0x_3^1+\binom2{1,1,0}x_1^1x_2^1x_3^0\;.
\end{align*}$$
