<s>[INST] Find a closed-form expression for $\binom n1+3\binom n3+5\binom n5+\cdots ,$ where $n > 1$. Find a closed-form expression for
$$\binom n1+3\binom n3+5\binom n5+\cdots ,$$
where $n > 1$. You may find the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$ helpful.

I got $2^{n-2}$ but it was wrong!  I don't know where I miscalculated...
 [/INST] Consider
$$(1+x)^n=1+\binom{n}{1}x+\binom{n}{2}x^2+\binom{n}{3}x^3+\binom{n}{4}x^4+\cdots$$
Differentiating with respect to $x$,
$$n(1+x)^{n-1}=\binom{n}{1}+2x\binom{n}{2}+3x^2\binom{n}{3}+\cdots$$
Put $x=1$,
$$n2^{n-1}=\binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}+\cdots\tag1$$
Put $x=-1$,
$$0=\binom{n}{1}-2\binom{n}{2}+3\binom{n}{3}-+\cdots\tag2$$
Taking $\frac{(1)+(2)}{2}$,
$$\binom{n}{1}+3\binom{n}{3}+5\binom{n}{5}+\cdots=n2^{n-2}$$
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