Bernoulli Triangle

Author

Eric W. Weisstein
December 9, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/BernoulliTriangle.nb.

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

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

Enumeration

ColumnForm[Table[Sum[Binomial[n,i],{i,0,k}],{n,0,6},{k,0,n}],Center]

Binary Plot

<<Utilities`Plot`

Length[t=Flatten[Table[Table[Sum[Binomial[n,i],{i,0,k}],{n,0,35},{k,0,n}]]]]

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

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

Analytic

f=Simplify[FunctionExpand[Sum[Binomial[n,i],{i,0,k}]]]

FullSimplify[n Gamma[n]]

f/.k->0

f/.k->n