Figurate Number Triangle

Author

Eric W. Weisstein
August 10, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sums

(∑_(j=0)^i)(Binomial[i, j])==(2^i)

Sum[Binomial[i,j],{j,0,i}]

(∑_(j=1)^i)(Binomial[i, j])==(2^i)-1

Sum[Binomial[i,j],{j,1,i}]

(∑_(i=0)^n)(Binomial[i, j])==((n+1)/(j+1)) (Binomial[n, j])

Sum[Binomial[i,j],{i,0,n}]

FullSimplify[%]

Table[{Sum[Binomial[i,j],{i,0,n}],((n+1)/(j+1))Binomial[n,j],((j Binomial[0,j]+(1-j+n) Binomial[1+n,j])/(1+j))},{j,0,5},{n,0,5}]