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This summation is just the sum of odd index binomial coefficients.  We know that the sum of the binomial coefficients from (0,n)=2^n $\sum_{i=0}^{n} {{n}\choose{i}} =2^n$  We also know that the alternation sum of binomial coefficients from (0,n)=0 (which is summation (i=0,n) of (-1)^i * nCi)  If you subtract the second equation from the first you get   2*summation from (i=0,n) of nC(2i+1)=2^n