Klee's Identity

Author

Eric W. Weisstein
October 5, 2008

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

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

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

Sum

(∑_(k=0)^n)(((-1))^k) (Binomial[n, k]) (Binomial[k+n, m])==(((-1))^n) (Binomial[n, m-n])

V6

Sum[(-1)^k Binomial[n,k]Binomial[n+k,m],{k,0,n}]

Table[-((Gamma[1+n] Gamma[-m+n] Sin[m π])/(π Gamma[1-m+2 n])),{m,0,10},{n,0,10}]

V7

Sum[(-1)^k Binomial[n,k]Binomial[n+k,m],{k,0,n}]

Table

TableForm[Table[Sum[(-1)^k Binomial[n,k]Binomial[n+k,m],{k,0,n}],{m,0,10},{n,0,10}],
TableAlignments->Right,
TableSpacing->1]

TableForm[Table[(-1)^n Binomial[n,m-n],{m,0,10},{n,0,10}],TableAlignments->Right,
TableSpacing->1]

Table[(-1)^n Binomial[n,m-n],{m,0,10},{n,Ceiling[m/2],m}]//ColumnForm

Recurrence

<<zb_alg.m

Zb[(-1)^k Binomial[n,k]Binomial[n+k,m],k,n,1][[1]]/.{F[k_,n_]:>a[n],Delta[__]:>0}//FullSimplify/.a[n_]:>Subscript[a,n]//TraditionalForm