% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly
% declare the paper format.
\usepackage[T1]{fontenc} % Not always necessary, but recommended.
\usepackage[utf8]{inputenc} % Not always necessary, but recommended.
% End of standard header. What follows pertains to the problem at hand.
\usepackage{mathtools}
\usepackage{tridelim}
\DeclarePairedDelimiter{\tripleparen}{\lPparen}{\rPparen}
\DeclarePairedDelimiter{\triplebrack}{\lBbrack}{\rBbrack}
\DeclarePairedDelimiter{\triplefloor}{\lFfloor}{\rFfloor}
\DeclarePairedDelimiter{\tripleceil} {\lCceil} {\rCceil}
\DeclarePairedDelimiter{\triplebrace}{\lBbrace}{\rBbrace}
\DeclarePairedDelimiter{\tripleangle}{\lAangle}{\rAangle}
\DeclarePairedDelimiter{\triplevert} {\lVvert} {\rVvert}
\DeclarePairedDelimiter{\triplegroup}{\lGgroup}{\rGgroup}
\DeclarePairedDelimiter{\triplearrowvert}{\Aarrowvert}{\Aarrowvert}
\DeclarePairedDelimiter{\triplebracevert}{\Bbracevert}{\Bbracevert}
\DeclarePairedDelimiterX\triplebracket[3]{\lAangle}{\rAangle}
{#1\,\delimsize\Vvert\,\mathopen{}#2\,\delimsize\Vvert\,\mathopen{}#3}
\newcommand*{\testfraction}{\frac{a}{b}}
\newcommand*{\testbigfraction}{%
\frac{\,\frac{d^{n}}{dt^{n}}\,}{n!}%
}
\newcommand*{\testmatrix}{%
\begin{matrix}%
1&0&0&0\\%
0&1&0&0\\%
0&0&1&0\\%
0&0&0&1%
\end{matrix}%
}
\begin{document}
We begin with a simple formula: \( P\lPparen x\rPparen \).
Is it OK in sub-\slash superscripts too? Let’s check:
\( Z^{P\lPparen x\rPparen} \), and \( 2^{Z_{P\lPparen x\rPparen}} \).
It works, but is suboptimal: indeed, compare it closely with
\( Z^{P(x)} \) and \( 2^{Z_{P(x)}} \).
Here are the normal sizes, in display:
\begin{align*}
& x \tripleparen{y} z &
& x \triplebrack{y} z &
& x \triplefloor{y} z &
& x \tripleceil {y} z &
& x \triplebrace{y} z
\\
& x \tripleangle{y} z &
& x \triplevert {y} z &
& y \triplearrowvert*{x} z & % needs "\left" and "\right"
& y \triplebracevert*{x} z % needs "\left" and "\right"
\end{align*}
Here are the \verb|\big| sizes, in display:
\begin{align*}
& x \tripleparen[\big]{y} z &
& x \triplebrack[\big]{y} z &
& x \triplefloor[\big]{y} z &
& x \tripleceil [\big]{y} z &
& x \triplebrace[\big]{y} z
\\
& x \tripleangle[\big]{y} z &
& x \triplevert [\big]{y} z &
& x \triplearrowvert[\big]{y} z &
& x \triplebracevert[\big]{y} z
\end{align*}
Here are the \verb|\Big| sizes, in display:
\begin{align*}
& x \tripleparen[\Big]{\testfraction} z &
& x \triplebrack[\Big]{\testfraction} z &
& x \triplefloor[\Big]{\testfraction} z &
& x \tripleceil [\Big]{\testfraction} z &
& x \triplebrace[\Big]{\testfraction} z
\\
& x \tripleangle[\Big]{\testfraction} z &
& x \triplegroup[\Big]{\testfraction} z & % not available at smaller sizes
& x \triplevert [\Big]{\testfraction} z &
& x \triplearrowvert[\Big]{\testfraction} z &
& x \triplebracevert[\Big]{\testfraction} z
\end{align*}
Here are the \verb|\bigg| sizes, in display:
\begin{align*}
& x \tripleparen[\bigg]{\testfraction} z &
& x \triplebrack[\bigg]{\testfraction} z &
& x \triplefloor[\bigg]{\testfraction} z &
& x \tripleceil [\bigg]{\testfraction} z &
& x \triplebrace[\bigg]{\testfraction} z
\\
& x \tripleangle[\bigg]{\testfraction} z &
& x \triplegroup[\bigg]{\testfraction} z &
& x \triplevert [\bigg]{\testfraction} z &
& x \triplearrowvert[\bigg]{\testfraction} z &
& x \triplebracevert[\bigg]{\testfraction} z
\end{align*}
Here are the \verb|\Bigg| sizes, in display:
\begin{align*}
& x \tripleparen[\Bigg]{\testbigfraction} z &
& x \triplebrack[\Bigg]{\testbigfraction} z &
& x \triplefloor[\Bigg]{\testbigfraction} z &
& x \tripleceil [\Bigg]{\testbigfraction} z
\\
& x \triplebrace[\Bigg]{\testbigfraction} z &
& x \tripleangle[\Bigg]{\testbigfraction} z &
& x \triplegroup[\Bigg]{\testbigfraction} z
\\
& x \triplevert [\Bigg]{\testbigfraction} z &
& x \triplearrowvert[\Bigg]{\testbigfraction} z &
& x \triplebracevert[\Bigg]{\testbigfraction} z
\end{align*}
And finally, the glorious auto-sizing delimiters, always in display:
\begin{align*}
& x \tripleparen*{\testmatrix} z &
& x \triplebrack*{\testmatrix} z &
& x \triplefloor*{\testmatrix} z
\\
& x \tripleceil* {\testmatrix} z &
& x \triplebrace*{\testmatrix} z &
& x \triplegroup*{\testmatrix} z
\\
& x \triplevert* {\testmatrix} z &
& x \triplearrowvert*{\testmatrix} z &
& x \triplebracevert*{\testmatrix} z
\end{align*}
For the following example, cf.~p.~26 of the manual of the \textsf{mathtools}
package:
\( \triplebracket[\big]{B}{\sum_{k} f_k}{C} \).
\end{document}
\Bigl\llbracket\!\Bigl[
or some suchnath
package, they are input as\triple[
and\triple]
, respectively; note, however, that this package also introduces a different way of writing math formulae in the source, making it superfuous, among other thigs, to use explicit\left
and\right
specifiers.