TITLE: ${n\choose1}+3{n\choose3}+5{n\choose5}+...$ in closed form
QUESTION [5 upvotes]: How can I evaluate ${n\choose1}+3{n\choose3}+5{n\choose5}+...$ in closed form?
Binomial theorem?  Is that what I'm suppose to use?  I'm not really understanding this.

REPLY [10 votes]: By the Binomial Theorem, we have 
$$\frac{(1+x)^n -(1-x)^n}{2}=\binom{n}{1}x +\binom{n}{3}x^3+\binom{n}{5}x^5+\cdots .$$
Differentiate with respect to $x$, and set $x=1$.