Aligning defined functions on sums

A stackengine approach, with scalerel for handling the change in math style size.

REVISED ANSWER

This can present nicely in all math styles and corrects the issue with my alternate solutions that the drop of a subscript is not uniform but depends on the letter height. For this application, that is not a helpful feature.

\documentclass{article}
\usepackage{stackengine,scalerel}
\stackMath
\newcommand\hypop[4]{%
  \ThisStyle{\litlet{#2}\stackengine{.2\LMex}{%
   \stackengine{.2\LMex}{\SavedStyle\triangle}{\scalebox{.5}{$\SavedStyle#4$}}{O}{c}{F}{F}{L}%
  }{\SavedStyle\litlet{#1}}{O}{c}{F}{F}{S}\litlet{#3}}%
}
\newcommand\litlet[1]{\raisebox{-.4\LMex}{\scalebox{.7}{$\SavedStyle#1$}}}
\begin{document}
\[
\hypop{a}{b}{c}{n} ~
\scriptstyle\hypop{a}{b}{c}{n}~
\scriptscriptstyle\hypop{a}{b}{c}{n}
\]
\[\sum^{\hypop{a}{b}{c}{n}}_{\hypop{a}{b}{c}{n}} \]
\end{document}

ORIGINAL ANSWER

The triangle is always presented in \displaystyle and argument #4 is always presented in \scriptscriptstyle. However, the point letters of the triangle (arguments #1-#3) get smaller when \hypop is used as a subscript.

\documentclass{article}
\usepackage{stackengine,scalerel}
\stackMath
\newcommand\hypop[4]{%
  \ThisStyle{{}_{#2} \stackengine{1pt}{%
   \stackengine{\dimexpr.72ex+.10\LMex}{\triangle}{_{_{\mkern2mu#4}}}{O}{c}{F}{F}{L}%
  }{\SavedStyle_{#1}}{O}{c}{F}{F}{S} {}_{#3}}%
}
\begin{document}
\[Intended~output~above~and~below~the~sum:\hypop{a}{b}{c}{n}\]
\[Actual~output:\sum^{\hypop{a}{b}{c}{n}}_{\hypop{a}{b}{c}{n}} \]
\end{document}

One can keep the triangle size fixed, but if a smaller size is preferred, a \scalebox would work, here scaled by 80% with respect to the original presentation above:

\documentclass{article}
\usepackage{stackengine,scalerel}
\stackMath
\newcommand\hypop[4]{%
  \ThisStyle{%
  {}_{#2} \stackengine{1pt}{\scalebox{.8}{%
   $\stackengine{\the\dimexpr.72ex+.10\LMex}{\triangle}{_{_{\mkern2mu#4}}}{O}{c}{F}{F}{L}$}%
  }{\SavedStyle_{#1}}{O}{c}{F}{F}{S} {}_{#3}}%
}
\begin{document}
\[Intended~output~above~and~below~the~sum:\hypop{a}{b}{c}{n}\]
\[Actual~output:\sum^{\hypop{a}{b}{c}{n}}_{\hypop{a}{b}{c}{n}} \]
\end{document}

Here's a version that uses a few additional \raisebox and \scalebox instructions to place the letters "b" and "c".

\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\newcommand{\ca}[1]{\operatornamewithlimits{\mathpalette\cb{#1}}}
\newcommand{\cb}[2]{{%
  \vphantom{\triangle}%
  \ooalign{$#1\triangle$\cr\hidewidth\cc{$#1#2$}\hidewidth\cr
  }%
}}
\newcommand{\cc}[1]{%
  \raisebox{.35\height}{\scalebox{0.45}{#1}}%
}
\newcommand{\hypop}[4]{%
    \raisebox{-0.25ex}{\scalebox{0.5}{$#2$}}
    \mkern-2.5mu
    \ca{#4}\limits^{\mkern-0.5mu\scalebox{0.5}{$#1$}}
    \mkern-2.8mu
    \raisebox{-0.25ex}{\scalebox{0.5}{$#3$}}}
\begin{document}
Various math styles:
$ \hypop{a}{b}{c}{n} \quad
  \scriptstyle  \hypop{a}{b}{c}{n} \quad 
  \scriptscriptstyle \hypop{a}{b}{c}{n}$

With summation symbol: 
$\displaystyle \sum^{\hypop{a}{b}{c}{n}}_{\hypop{a}{b}{c}{n}}$
\end{document}