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A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.

%I A262666 #39 Nov 12 2015 04:42:45
%S A262666 1,1,1,1,0,2,0,1,1,1,2,2,2,2,2,2,1,1,1,0,4,0,8,0,12,0,14,0,12,0,8,0,4,
%T A262666 0,1,1,1,4,4,10,10,20,20,31,31,40,40,44,44,40,40,31,31,20,20,10,10,4,
%U A262666 4,1,1,1,0,6,0,21,0,56,0,120,0,216,0,336,0,456,0
%N A262666 Irregular table read by rows: T(n,k) is the number of binary bisymmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2.
%C A262666 T(n,k) = 0 if n is even and k is odd.
%C A262666 T(n,k) = T(n,k+1) if n is odd and k is even.
%H A262666 Alois P. Heinz, <a href="/A262666/b262666.txt">Rows n = 0..32, flattened</a>
%H A262666 Kival Ngaokrajang, <a href="/A262666/a262666_1.pdf">Illustration of initial terms</a>
%H A262666 Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/BINRMAT.pdf">Notes on binary bisymmetric matrices</a>
%H A262666 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bisymmetric_matrix">Bisymmetric Matrix</a>
%F A262666 G.f. for row n: (1+x)^t*(1+x^2)^(n-t)*(1+x^4)^(((n-2)*n+t)/4) where t = n mod 2. - _Alois P. Heinz_, Sep 27 2015
%e A262666 Irregular table begins:
%e A262666 n\k 0   1   2   3   4   5   6   7   8   9   ...
%e A262666 0:  1
%e A262666 1:  1   1
%e A262666 2:  1   0   2   0   1
%e A262666 3:  1   1   2   2   2   2   2   2   1   1
%e A262666 4:  1   0   4   0   8   0  12   0  14   0   ...
%e A262666 5:  1   1   4   4  10  10  20  20  31  31   ...
%e A262666 ...
%p A262666 T:= n-> seq(coeff((t->(1+x^2)^(n-t)*(1+x)^t*(1+x^4)^
%p A262666       (((n-2)*n+t)/4))(irem(n, 2)), x, i), i=0..n^2):
%p A262666 seq(T(n), n=0..6);  # _Alois P. Heinz_, Sep 27 2015
%Y A262666 Row sums give A060656(n+1).
%Y A262666 Columns k=0-3 give: A000012, A000035, A052928, A237420(n+1).
%Y A262666 Cf. A184948, A261242.
%K A262666 nonn,tabf
%O A262666 0,6
%A A262666 _Kival Ngaokrajang_, Sep 26 2015
%E A262666 More terms from _Alois P. Heinz_, Sep 27 2015