3-ary Boolean functions; balanced families

In the triangle T = the entry T(n,k) shows the number of  n-ary families that contain 2k functions.

Row n=3 has the entries 0,7,0,7.

T(3,1) = 7
The seven 3-ary families that contain 21 = 2 functions: These are linear functions (tribe E0). They correspond to the rows of a Walsh matrix, except row 0. (The corresponding box in this cube is empty.)

T(3,3) = 7
The seven 3-ary families that contain 23 = 8 functions: