Property of Binomial Coefficient I have a "must be trivial" problem which I could not solve. 
Prove the following relation of binomial coefficients, if true:
$$\sum_{k=1}^{n}{2n+1 \choose k}=2^{2n}-1$$
P.S. Though this is not homework, I appreciate any hints rather than explicit solutions. 
I am looking for proofs from properties of binomial coefficients rather than other methods.

Substitute $x=1$ into $(1+x)^{2n+1}=\sum_{k=0}^{2n+1}\binom{2n+1}{k}x^k$ to obtain  $2^{2n+1}=\sum_{k=0}^{2n+1}\binom{2n+1}{k}=2\sum_{k=0}^{n}\binom{2n+1}{k}$ since $\binom{2n+1}{k}=\binom{2n+1}{2n+1-k}$.
