Evaluate the sum ${n \choose 1} + 3{n \choose 3} +5{n \choose 5} + 7{n \choose 7}...$ in closed form How do I evaluate the sum:$${n \choose 1} + 3{n \choose 3} +5{n \choose 5} + 7{n \choose 7}  ...$$in closed form?
I don't really know how to start and approach this question. Any help is greatly appreciated.

Starting with 
\begin{align}
(1+t)^{n} = \sum_{k=0}^{n} \binom{n}{k} t^{k}
\end{align}
it is seen that
\begin{align}
\frac{1}{2} \left[ (1+t)^{n} - (1-t)^{n} \right] = \sum_{k=1}^{[(n+1)/2]} \binom{n}{2k-1} t^{2k-1}.
\end{align}
Now differentiating both sides leads to
\begin{align}
\sum_{k=1}^{[(n+1)/2]} (2k-1) \binom{n}{2k-1} t^{2k-2} = \frac{n}{2} \left[ (1+t)^{n-1} + (1-t)^{n-1} \right]
\end{align}
and upon setting $t=1$ the desired result is
\begin{align}
\sum_{k=1}^{[(n+1)/2]} (2k-1) \binom{n}{2k-1} = 2^{n-2} \ n.
\end{align}  
