#[05AB1E](https://github.com/Adriandmen/05AB1E/wiki/Commands), 3 [bytes](https://github.com/Adriandmen/05AB1E/wiki/Codepage)

    Ýo&

Port of [*@JonathanAllan*'s Jelly answer](https://codegolf.stackexchange.com/a/179178/52210), so make sure to upvote him!

Contains zeros (including -loads of- trailing zeros).

[Try it online](https://tio.run/##yy9OTMpM/f//8Nx8tf//DU0B) or [verify all test cases](https://tio.run/##yy9OTMpM/V9WeXiCvZLCo7ZJCgpK9v8Pz81X@6/zP9pQR8FIR8HQFIjNdBTMTIAYyLYAso1MDGIB).

**Explanation:**

<!-- language-all: lang-python -->

    Ý      # Create a list in the range [0, (implicit) input]
           #  i.e. 15 → [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
           #  i.e. 16 → [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
     o     # Take each value to the power of 2
           #  → [1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768]
           #  → [1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536]
      &    # Bitwise-AND each value with the (implicit) input
           # 15 → [0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0]
           # 16 → [1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0]
           # (and output the result implicitly)