Help interpret notation $\sum_{j_1+j_2+\cdots+j_m=n}a^{j_1}_1a^{j_2}_2\cdots a_m^{j_m}$

Question

I read somewhere that this sum can be written as: $$\sum_{r+s=n}a_rb_s=\sum_{r=0}^na_rb_{n-r}\tag1$$ This means to create all possible orders of $(r,s)$ and add these together.

Now, my question is how do you write this summation in terms of the RHS above:

$$\sum_{j_1+j_2+\cdots+j_m=n}a^{j_1}_1a^{j_2}_2\cdots a_m^{j_m}=?\tag2$$

Answer 1

For $m=3$,

$$\sum_{r=0}^n\sum_{s=0}^{n-r}a_rb_s c_{n-r-s}.$$

For $m=4$,

$$\sum_{r=0}^n\sum_{s=0}^{n-r}\sum_{t=0}^{n-r-s}a_rb_s c_td_{n-r-s-t}.$$

And so on.