Is there a common notation to denote the developped form as double sum of
$$ (A+B)^n = \sum_{k=0}^n \sum \ldots $$ With se second sum having $\binom{n}{k}$ terms in it.
For example with $n=3$
$$ (A+B)^3 = A^3 + \underbrace{(AAB + ABA + BAA)}_{\binom{3}{1} \text{ terms}} + \underbrace{(ABB + BAB + BBA)}_{\binom{3}{2} \text{ terms}} + B^3 $$
I assume no one encountered notation for this, so I'll share the one I'm using, that isn't so bad. I write $$ \underline{A^k B^{n-k}} = \text{the sums of all the different order of product for $k$ $A$'s and $n-k$ $B$'s} $$ For example $$ \underline{A^2 B} = \underline{ABA} = \underline{BA^2} = A^2 B + ABA + B A^2 $$ Thus we can write $$ (A+B)^n = \sum_{k=0}^n \underline{A^k B^{n-k}} $$ The notation carry this property I needed $$ \underline{(xA)^k (yB)^{n-k}} = x^k y^{n-k} \underline{A^k B^{n-k}} $$