N numbers closest to zero staying balanced

Pyth, 10 bytes

*-R/Q2Q%Q2

Try it online.

How it works

            (implicit) Store the input in Q.
   /Q2      Calculate Q/2 (integer division).
 -R   Q     Subtract that value (-R) from each element in [0, ..., Q-1] (Q).
*      %Q2  Repeat the resulting array Q%2 times.

Mathematica, 32 30 24 bytes

OddQ@#&&Range@#-(#+1)/2&

Code-golfing trick: The last argument to And doesn't have to be a boolean.

APL, 16 15 13 bytes

Thanks to @Dennis for -2 bytes!

⌊(⍳⊢×2|⊢)-÷∘2

This is a monadic train that gives an empty array for even input. Below is the diagram:

┌────┴─────┐   
⌊ ┌────────┼─┐ 
┌─┴─┐      - ∘ 
⍳ ┌─┼───┐   ┌┴┐
  ⊢ × ┌─┼─┐ ÷ 2
      2 | ⊢    

First, ⊢×2|⊢ gives the input times its mod 2; that is, odds will give themselves, and evens give 0. We use ⍳ to create a list of numbers from 1 to that (⍳0 gives the empty array), and then we subtract half the input and floor.