makeatletter
 newcommandbinomialCoefficient[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%
     advancec@pgf@countc by-c@pgf@countb%
     ifnumc@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
         ifnumc@pgf@countd<c@pgf@countb%
         multiplyc@pgf@countc byc@pgf@counta%
         advancec@pgf@counta by-1%
         advancec@pgf@countd by1%
         dividec@pgf@countc byc@pgf@countd%
     repeatpgfmathloop%
     thec@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}