<s>[INST] Is there a name for a binomial expansion without coefficients? I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here:
Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
I was led astray for a bit, because I misunderstood the rhs to be the difference of $x$ and $y$ times the binomial expansion $(x+y)^{n-1}$.
But this is wrong... the rhs is actually the difference of $x$ and $y$ times the binomial expansion $(x+y)^{n-1}$ "without the coeffients", i.e., 
$$\sum_{i=0}^{n-1}x^{n-1-i}y^i$$
(also briefly discussed here: Binomial summation without coefficient)
Is there a name for this summation?
 [/INST] 
There is no specific name for the bivariate polynomial
  \begin{align*}
x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1}
\end{align*}
but the nice expression
  \begin{align*}
(x-&y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1})\\
\end{align*}
is called a telescoping sum since all terms besides the first and the last cancel out.
\begin{align*}
(x-&y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1})\\
&=x(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1})\\
&\qquad-y(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1})\\
&=\color{blue}{x^{n}}+x^{n-1}y+x^{n-2}y^2+\cdots+x^3y^{n-3}+x^2y^{n-2}+xy^{n-1}\\
&\qquad\ \;-x^{n-1}y-x^{n-2}y^2-\cdots-x^3y^{n-3}-x^2y^{n-2}-xy^{n-1}\color{blue}{-y^{n}}\\
&=\color{blue}{x^n-y^n}\\
\end{align*}

</s>