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%I #39 Nov 02 2022 07:41:43
%S 0,1,0,3,2,1,0,15,10,5,12,3,6,9,8,7,2,13,4,11,14,1,0,255,170,85,204,
%T 51,102,153,136,119,34,221,68,187,238,17,240,15,90,165,60,195,150,105,
%U 120,135,210,45,180,75,30,225,160,95,10,245,108,147
%N Table of binary Walsh functions w(A001317), columns read as binary numbers.
%C The rows of an infinite binary Walsh matrix (compare A228539) are the binary Walsh functions w(0),w(1),w(2),w(3),...
%C This number triangle represents the infinite binary array w(1),w(3),w(5),w(15),... (1,3,5,15,... is A001317.)
%C T(n,k) is column k of the (2^n) X (2^2^n) submatrix read as a binary number.
%C Top left 4 X 16 submatrix of the binary array:
%C 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
%C 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0
%C 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0
%C 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
%C In the sequence this is represented by row 2:
%C 0 15 10 5 12 3 6 9 8 7 2 13 4 11 14 1
%C 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.
%C Each odd column is the complement of the even column on its left.
%C 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.
%H Tilman Piesk, <a href="/A197819/b197819.txt">Rows 0..3 of the triangle, flattened</a>
%H Tilman Piesk, permutation matrices of <a href="http://commons.wikimedia.org/wiki/File:Walsh_permutation_1_3_5_15.svg">row 2</a> and <a href="http://commons.wikimedia.org/wiki/File:Walsh_permutation_1_3_5_15_17_51_85_255.svg">row 3</a>
%H Tilman Piesk, <a href="/A195467/a195467_7.txt">Explanations</a> (including the 8x256 submatrix) and <a href="/A195467/a195467_5.txt">MATLAB code</a> showing the connection with A195467
%Y Cf. A195467 (consecutive powers of the Gray code permutation).
%Y Cf. A001317 (Sierpinski triangle rows read like binary numbers).
%K nonn,tabf
%O 0,4
%A _Tilman Piesk_, Oct 18 2011, reviewed Aug 25 2013
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