Inverse discrete Fourier transform
No it's not correct.

Derive it like this :

1-) $2N$-point DFT of $x[n]$ is: $X_{2N}[k] = \sum_{n=0}^{2N-1} x[n] e^{-j \frac{2\pi}{2N} k n }$

2-) $Y[k] = X_{2N}[2k+1]$ be the odd-indexed samples of $X_{2N}[k]$

3-) We are looking for $y[n] = \text{N-point IDFT}\{ Y[k]\}$.

4-) Elaborate on step-2 and step-1 to see that $Y[k] = \text{N-point DFT}\{ x[n] e^{-j \frac{\pi}{N}n}\}$

5-) Then from steps 3 and 4 we get :

$$y[n] = \text{N-point IDFT}\{ \text{N-point DFT}\{ x[n] e^{-j \frac{\pi}{N}n}\} \}  $$

$$ y[n] = x[n] e^{-j \frac{\pi}{N}n}  $$