Let $n$ be a positive integer which can be arbitrarily large.
Consider the "box" (I don't know the correct word) whose sides are of lengths $a_1,...a_n$, and where I'm interested in the case where the first $K$ sides have the same length, say $a$, and the other sides are consecutive onwards, i.e. $a_{K+1}=a+1$, $a_{K+2}=a+2$,...
I would like to express the volume of that box as a function of simpler objects, like linear combinations of "cubes" $b^n$, for well-chosen $b$'s.
Edit: for example is there a combinatorial way to describe in a common fashion the identities $a^3(a+1)(a+2)=a^5+3a^4+2a^3$ and $a^7(a+1)(a+2)(a+3)(a+4)=a^11 + 10a^10+35a^9+50a^8+24a^7$?
Is this well-studied? What is an expression for a solution, or what are relevant books, papers and keywords?