Ejemplo de sumas sobre sumas anidadas  (Samuel David López Armenta):

\documentclass{article}

\usepackage{amsmath, xcolor}

\begin{document}

\section{Definición Original (Texto Literal)}

\[

\mathcal{S}^{({\color{blue}k})}_n[f({\color{red}x})] = {\color{blue}\sum_{x=0}^n \sum_{x=0}^n \cdots \sum_{x=0}^n} f({\color{red}x}) = \frac{({\color{blue}k}-1 + n - {\color{red}x})!}{({\color{blue}k}-1)! \, (n - {\color{red}x})!} \cdot f({\color{red}x})

\]

\subsection*{Características Clave}

\begin{itemize}

    \item \textbf{Mismo índice ${\color{red}x}$ en todos los sumatorios}: Es una elección deliberada (no un error).

    \item \textbf{Resultado combinatorio}: Los coeficientes surgen de la estructura anidada, \textbf{no} de una analogía externa.

    \item \textbf{Propósito único}: No es equivalente a sumatorios estándar ni a potencias de sumas.

\end{itemize}

\end{document}

\mathcal{S}^{({\color{blue}4})}_n[f({\color{red}x})]={\color{blue}\sum_{x=0}^{n}\sum_{x=0}^{n}{\sum_{x=0}^{n}} \sum_{x=0}^{n}} f({\color{red}x})=\frac{(({\color{blue}4}-1)+(n-{\color{red}x}))!}{({\color{blue}4}-1)!(n-{\color{red}x})!}\cdot f({\color{red}x})=\frac{(({\color{blue}4}-1)+(3-{\color{red}0}))!}{({\color{blue}4}-1)!(3-{\color{red}0})!}\cdot f({\color{red}0})+\frac{(({\color{blue}4}-1)+(3-{\color{red}1}))!}{({\color{blue}4}-1)!(3-{\color{red}1})!}\cdot f({\color{red}1})+\frac{(({\color{blue}4}-1)+(3-{\color{red}2}))!}{({\color{blue}4}-1)!(3-{\color{red}2})!}\cdot f({\color{red}2})+\frac{(({\color{blue}4}-1)+(3-{\color{red}3}))!}{({\color{blue}4}-1)!(3-{\color{red}3})!}\cdot f({\color{red}3})