Euler's Number Triangle

Author

Eric W. Weisstein
December 9, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/EulersNumberTriangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EulersNumberTriangle.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`Combinatorica`
<<Utilities`Notation`

Notation

Notation[⟨({{n_}, {k_}})⟩ ⟺ eulerian[n_,k_]]

TableForm[Table[eulerian[n,k],{n,4},{k,0,n-1}],Row Alignments->Center]

Triangle

ColumnForm[Table[Eulerian[n,k],{n,6},{k,0,n-1}],Center]

Recurrnence

RSolve[{a[n+1,k]==k a[n,k]+(n+2-k)a[n,-1],a[1,0]==1,a[n,0]==1},a[n,k],{n,k}]

Binary Plot

<<Utilities`Plot`

Length[t=Flatten[Table[Table[Eulerian[n,k],{n,0,35},{k,0,n}]]]]

Show[PixelPerfect[BasePlot[Take[t,255]],BlockSize->2]]

Show[PixelPerfect[BasePlot[Take[t,511]],BlockSize->1]]

As Z-Transform

ZTransform[Table[n^k,{k,5}],n,z]

ColumnForm[Rest/@(CoefficientList[#,z]&/@Numerator/@ZTransform[Table[n^k,{k,6}],n,z]),Center]