Pascal's triangle in tikz

Question

I would like to typeset the top part of Pascal's triangle. To get the triangle with the names of the binomial coefficients, i.e., {n \choose k}, I used the following code

\begin{tikzpicture}
\foreach \n in {0,...,4} {
  \foreach \k in {0,...,\n} {
    \node at (\k-\n/2,-\n) {${\n \choose \k}$};
  }
}
\end{tikzpicture}

The result is this

Now I want to be equally lazy and do something like this for the values of the binomial coefficients, i.e., replace {\n \choose \k} in the node label with \CalculateBinomialCoefficient{\n}{\k} where \CalculateBinomialCoefficient is a hypothetical macro that calculates the binomial coefficient. Has anyone done something like that?

The result should look like this:

Answer 1

From texample.net. The author is Paul Gaborit.

Triangle de Pascal

Answer 2

Here is a solution using TeX integer arithmetic. I am reusing counters defined by PGF in order to avoid having to declare new ones.

\documentclass{article}
\usepackage{tikz}

\makeatletter
\newcommand\binomialCoefficient[2]{%
    % Store values 
    \c@pgf@counta=#1% n
    \c@pgf@countb=#2% k
    %
    % Take advantage of symmetry if k > n - k
    \c@pgf@countc=\c@pgf@counta%
    \advance\c@pgf@countc by-\c@pgf@countb%
    \ifnum\c@pgf@countb>\c@pgf@countc%
        \c@pgf@countb=\c@pgf@countc%
    \fi%
    %
    % Recursively compute the coefficients
    \c@pgf@countc=1% will hold the result
    \c@pgf@countd=0% counter
    \pgfmathloop% c -> c*(n-i)/(i+1) for i=0,...,k-1
        \ifnum\c@pgf@countd<\c@pgf@countb%
        \multiply\c@pgf@countc by\c@pgf@counta%
        \advance\c@pgf@counta by-1%
        \advance\c@pgf@countd by1%
        \divide\c@pgf@countc by\c@pgf@countd%
    \repeatpgfmathloop%
    \the\c@pgf@countc%
}
\makeatother

\begin{document} 
\begin{tikzpicture}
\foreach \n in {0,...,15} {
  \foreach \k in {0,...,\n} {
    \node at (\k-\n/2,-\n) {$\binomialCoefficient{\n}{\k}$};
  }
}
\end{tikzpicture}

\end{document}

If you want, you can wrap \pgfmathdeclarefunction around that to have the function available in pgfmath (see Section 65 “Customizing the Mathematical Engine” in the manual (v2.10)).