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The rows of an infinite binary Walsh matrix (compare A228539) are the binary Walsh functions w(0),w(1),w(2),w(3),...

This number triangle represents the infinite binary array w(1),w(3),w(5),w(15),... (1,3,5,15,... is A001317.)

T(n,k) is column k of the (2^n) X (2^2^n) submatrix read as a binary number.

Top left 4 X 16 submatrix of the binary array:

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0

0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

In the sequence this is represented by row 2:

0 15 10 5 12 3 6 9 8 7 2 13 4 11 14 1

A195467 is the infinite array of Gray code permutation powers. It can be defined by this binary array, which happens to be A195467 mod 2.

Each odd column is the complement of the even column on its left.

Each row of the number triangle is a self-inverse Walsh permutation. The subsequence of even entries (on the even places) divided by 2 is a self-inverse Walsh permutation too.

Conjectures: 1) there are only 3 different first differences 1,4,6; 2) the sequence contains either isolated series of terms, e.g., {1},{7},{25},{31},..., or series of 3 consecutive integers, e.g., {11,12,13}, {19,20,21}, etc.; 3)the first terms m of every series satisfy the condition A(m+1)-A(m-1)=5, where A(n)=A001969(n).