Legendre Transformation

Author

Eric W. Weisstein
October 22, 2008

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

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

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

Sums

(∑_(k=0)^n)((Binomial[n, k])^2) ((Binomial[k+n, k])^2)==(∑_(k=0)^n)(Binomial[n, k]) (Binomial[k+n, k]) (∑_(j=0)^k)((Binomial[k, j])^3)

Sum[((Binomial[n,k])^2)((Binomial[n+k,k])^2),{k,0,n}]

Sum[Binomial[n,k]Binomial[n+k,k]Sum[((Binomial[k,j])^3),{j,0,k}],{k,0,n}]

Subtract@@@Table[{Sum[((Binomial[n,k])^2)((Binomial[n+k,k])^2),{k,0,n}],Sum[Binomial[n,k]Binomial[n+k,k]Sum[((Binomial[k,j])^3),{j,0,k}],{k,0,n}]},{n,0,10}]//Union