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.