Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$

Question

Prove that every number to a power greater than 1 can be written as

$X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$

for all natural numbers $X,Y$ where $X>Y$.

Further, there exist integers $c,d,e,f$ that result in a unique solution when $a_{i},b_{i}$ are restricted to the ranges $c\leq a_{i}\leq d$ and $e \leq b_{i}\leq f$.

This is more general than a specific question I asked previously.