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%I #12 Oct 15 2015 17:05:11
%S 1,1,1,1,0,0,0,1,1,1,2,2,2,2,2,2,1,1,1,0,0,0,4,0,0,0,6,0,0,0,4,0,0,0,
%T 1,1,1,4,4,10,10,20,20,31,31,40,40,44,44,40,40,31,31,20,20,10,10,4,4,
%U 1,1,1,0,0,0,9,0,0,0,36,0,0,0,84,0,0,0,126,0,0
%N Irregular table read by rows: T(n,k) is the number of binary symmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2. Where the symmetry axes are in horizontal and vertical.
%C The row length of this irregular triangle is n^2+1 = A002522(n).
%C Inspired by A262666, but rotating the diagonal and antidiagonal symmetry axis to horizontal and vertical axes.
%C From _Wolfdieter Lang_, Oct 12 2015 (Start):
%C Double symmetry of n X n matrix M: M(i, j) = M(n-i+1, j) = M(i, n-j+1) (= M(n-i+1, n-j+1)), here with entries from {0, 1}.
%C Due to 0 <-> 1 flip the rows are symmetric.
%C The number of independent entries in such an n X n doubly symmetric matrix is A008794(n+1) (squares repeated). Therefore, the row sums give repeated A002416 (omitting the first 1): 1, 2, 2, 16, 16, 512, 512, ... (End) - _Wolfdieter Lang_, Oct 12 2015
%H Kival Ngaokrajang, <a href="/A262742/a262742.pdf">Illustration of initial terms</a>
%e Irregular table begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 ...
%e 0: 1
%e 1: 1 1
%e 2: 1 0 0 0 1
%e 3: 1 1 2 2 2 2 2 2 1 1
%e ...
%e Row 4: 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1;
%e Row 5: 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1.
%e ...
%Y Cf. A262666,A002522, A008794, A002416.
%K nonn,tabf
%O 0,11
%A _Kival Ngaokrajang_, Sep 29 2015
%E More terms from _Alois P. Heinz_, Sep 29 2015
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