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A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers.

%I A228539 #21 Oct 28 2021 10:00:47
%S A228539 0,0,2,0,10,12,6,0,170,204,102,240,90,60,150,0,43690,52428,26214,
%T A228539 61680,23130,15420,38550,65280,21930,13260,39270,4080,42330,49980,
%U A228539 27030,0,2863311530,3435973836,1717986918,4042322160,1515870810,1010580540
%N A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers.
%C A228539 T(n,k) is row k of the binary Walsh matrix of size 2^n read as reverse binary number. The left digit is always 0, so all entries are even.
%C A228539 Most of these numbers are divisible by Fermat numbers (A000215): All entries in all rows beginning with row n are divisible by F_(n-1), except the entries 2^(n-1)...2^n-1. (This is the same in A228540.)
%C A228539 Divisibility by Fermat numbers:
%C A228539 All entries are divisible by F_0 = 3, except those with k = 1.
%C A228539 All entries in rows n >= 3 are divisible by F_2 = 17, except those with k = 4..7.
%H A228539 Tilman Piesk, <a href="/A228539/b228539.txt">Rows 0..8 of the triangle, flattened</a>
%H A228539 Tilman Piesk, <a href="/A228539/a228539.txt">Prime factorizations</a>
%H A228539 Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Binary_Walsh_matrix_256.svg">Binary Walsh matrix of size 256</a>
%F A228539 T(n,k) + A228540(n,k) = 2^2^n - 1
%F A228539 T(n,2^n-1) = A122570(n+1)
%e A228539 Binary Walsh matrix of size 4 and row 2 of the triangle:
%e A228539 0 0 0 0         0
%e A228539 0 1 0 1        10
%e A228539 0 0 1 1        12
%e A228539 0 1 1 0         6
%e A228539 Triangle starts:
%e A228539    k  =  0     1     2     3     4     5     6     7     8     9    10    11 ...
%e A228539 n
%e A228539 0        0
%e A228539 1        0     2
%e A228539 2        0    10    12     6
%e A228539 3        0   170   204   102   240    90    60   150
%e A228539 4        0 43690 52428 26214 61680 23130 15420 38550 65280 21930 13260 39270 ...
%Y A228539 Cf. A228540 (the same for the negated binary Walsh matrix).
%Y A228539 Cf. A000215 (Fermat numbers), A023394 (Prime factors of Fermat numbers).
%K A228539 nonn,tabf
%O A228539 0,3
%A A228539 _Tilman Piesk_, Aug 24 2013