  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.
