If $X(k)$ is the $N$-point DFT of $x(n)$, and $y(n)= x\left(\frac {n+1}{2}\right)$ for odd $n$, and $0$ for even $n$.
What is the $2N$ point DFT of $y(n)$ in terms of $X(k)$?
So far, I've noticed that $y(2n+1)=x(n+1)$, but I'm not sure what to do with that information in terms of the DFT summation. Can someone help me out here?
I would follow a signal block diagram based solution for this problem.
First as suggested in the comments, it's very helpful to investigate a few values of $y[n]$ and $x[n]$ for some $n$ :
$$ \begin{align} y[0] &= 0 ~~~,~~~ y[1] = x[1] \\ y[2] &= 0 ~~~,~~~ y[3] = x[2] \\ y[4] &= 0 ~~~,~~~ y[5] = x[3] \\ y[6] &= 0 ~~~,~~~ y[7] = x[4] \\ \end{align} $$
with some experience, or by a direct investigation of this above sequence it can be seen that the system that produce $y[n]$ from $x[n]$ is the folowing :
$$ x[n] \longrightarrow \boxed{ \uparrow 2} \longrightarrow w[n] \longrightarrow \boxed{ z^1} \longrightarrow y[n]$$
where the up arrow indicates an expansion by $2$ and the $z^1$ indicates a left shift (advance) by one sample.
Write down the DTFT relations between those signals $x[n]$,$w[n]$ and $y[n]$ :
$$ \begin{align} W(\omega) &= X(2\omega) \\ Y(\omega) &= e^{j\omega} W(\omega) \\ \\ Y(\omega) &= e^{j\omega} X(2\omega) \\ \end{align} $$
And relate $2N$ point DFT $Y[k]$ of $y[n]$ to $N$ point DFT $X[k]$ of $x[n]$.
$$ \begin{align} Y[k] &= Y(\omega_k) = Y(\frac{2\pi}{2N} k) &, ~~ k=0,1,...,2N-1\\ Y[k] &= e^{j \frac{2\pi}{2N} k} X(\frac{2\pi}{N} k) &, ~~ k = 0,1,...,2N-1\\ Y[k] &= e^{j \frac{\pi}{N} k} X[k] &, ~~ k = 0,1,...,2N-1 \end{align} $$
It can be seen that the length $2N$ DFT sequence $Y[k]$ consists of two fold replica of length $N$ DFT sequence $X[k]$ weighted by $e^{j \frac{\pi}{N} k}$. Note that $X[k]$ is periodic in $k$ by $N$, hence repeats twice during $k = 0,1,...,2N-1$.
