An unusual identity for odd-powers

In this manuscript we provide a new polynomial pattern. This pattern allows to find a polynomial expansion of the form $$x^{2m+1}=\sum _{k=1}^x\sum _{r=0}^m{\mathbf{A}}_{m,r}{k}^r(x-k{)}^r,$$ where $x,m\in \mathbb{N}$ and ${\mathbf{A}}_{m,r}$ is real coefficient.