If you paint the six sides of a 3x3x3 cube and then slice it into 27 unit cubes, how many will have paint on 0, 1, 2, 3 of their square sides?

Generalize to any dimension:
If you paint the sides of a n dimensional cube with sides length 3 and slice it into 3ⁿ unit cubes, how many will have paint on 0, 1, 2, ..., n of their (n-1) dimensional sides?