Could the sequence A287326 be generalized in order to receive expansion of natural power n>3?

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: Overview of preprint 1603.02468
• Sequence A287326: https://oeis.org/A287326
• Dedicated preprint: Series Representation of Power Function
• 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

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