<s>[INST] Sum of combinatorics sequence $\binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1}$ I need to find sum like  $$\binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1},\qquad \text{ for even } n$$
Example:
Find the sum of $$\binom{20}{1} + \binom{20}{3} +\cdots+ \binom{20}{19}=\ ?$$
 [/INST] Hint: Use the fact that ${n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}$. Since $n$ is even, let $n = 2p$ for some $p$. Then we have:
$$\sum_{r\ \text{ odd}}{n \choose r} = \sum_{i=1}^p{2p \choose 2i-1} = \sum_{i=1}^p{2p-1 \choose 2i-2}+{2p-1 \choose 2i-1} = \sum_{i=0}^{2p-1}{2p-1 \choose i}$$
This may be easier to see with a specific example:
$${20 \choose 1} + {20 \choose 3} + \ldots + {20 \choose 19} = \left[{19 \choose 0} + {19 \choose 1}\right] + \left[{19 \choose 2} + {19 \choose 3}\right] + \left[{19 \choose 18} + {19 \choose 19}\right]$$
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