The sequence https://oeis.org/A287326 - is Binomial distributed triangular array, that shows us necessary items to expand perfect cube $n^3$. Summation of $n$-th row of Triangle A287326 from $0$ to $n-1$ returns $n^3$. But is it exist simillar patterns in order to receive expansion of power $n>3$, where $n$ - positive integer?

$$
    \begin{matrix}
    &    &    &    &    &  1\\
&    &    &    &  1 &   &  1\\
&    &    &  1 &    &  7&    &  1\\
&    &  1 &    &  13&   &  13&    &  1\\
&  1 &    &  19&    & 25&    &  19&    &  1\\
    \end{matrix}
$$
Figure 1. Triangle A287326.

It derived by means of identity 
$$
x^3=\sum\limits_{m=0}^{x-1}3!\cdot mx-3!\cdot m^2+1
$$

For detailed info on derivation, please, reffer to links below. Thank you !

 - Derivation of A287326: https://kolosovpetro.github.io/pdf/Overview_of_preprint_1603.02468.pdf
 - Sequence A287326: https://oeis.org/A287326
 - Dedicated preprint: https://kolosovpetro.github.io/pdf/series_representation_of_power_function.pdf
 - Related preprints: https://kolosovpetro.github.io/
 - This question is a part of RG project: https://www.researchgate.net/project/Research-on-Binomial-Theorem-and-sequence-A287326-in-OEIS