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Verifying that a symbolic expression in two variables is 0.

I want to verify using SAGE that the expression:

${~~n~+1\choose k}~2^{-n~-1} - {~n\choose k}~2^{-n} + {~n\choose ~k~}~2^{-n~-1} + {~n \choose ~k-1}~2^{-n~~-1}$

is identically zero.

The SAGE script for this is:

binomial(n+1,k)2^(-n-1) - binomial(n,k)2^(-n) + binomial(n,k)2^(-n-1) + binomial(n,k-1)2^(-n-1)

I have tried using full_simplify() but it doesn't boil the expression down to zero.

Please help.