3-ary Boolean functions; balanced families

3-ary Boolean functions

In the triangle T =  A227724 the entry T(n,k) shows the number of balanced n-ary families that contain 2^k 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:

		tribe E2		tribe E3
	tribe E1		tribe E2
		tribe E1		tribe E2
			tribe E1