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%I #8 Mar 12 2014 16:37:03
%S -1,-1,-1,2,2,1,1,1,1,-1,1,0,-1,2,1,2,0,0,4,1,-1,-1,0,-1,-6,-4,2,1,1,
%T 1,-1,-7,-3,7,1,-1,-2,-2,-1,8,6,-12,-7,2,1,-1,-1,-2,10,3,-23,-6,10,1,
%U -1,-1,0,2,-12,-7,34,22,-18,-10,2,1,-2,0,0,-16,-4,52,16,-48,-9,13,1,-1,1,0,2,20,13,-70,-46,78,47,-24,-13,2,1,-1,-1,2,22,9
%N Tri-antidiagonal matrices of central ones with upper negative one to give a triangular sequence: first element is negative one. k=1;m(n,m,d)=If[n + m - 1 == d && n > 1, k, If[n + m ==d, -1, If[n + m - 2 == d, -1, If[n == 1 && m == d, -k, 0]]]].
%C Matrices: 1 X 1 {{-1}}, 2 X 2 {{-1, -1}, {1, -1}}, 3 X 3 {{0, -1, -1}, {-1, 1, -1}, {1, -1, 0}}, 4 X 4 {{0, 0, -1, -1}, {0, -1, 1, -1}, {-1, 1, -1, 0}, {1, -1, 0, 0}}, 5 X 5 {{0,0, 0, -1, -1}, {0, 0, -1, 1, -1}, {0, -1, 1, -1, 0}, {-1, 1, -1, 0, 0}, {1, -1, 0, 0, 0}}
%F k=1; m(n,m,d)=If[n + m - 1 == d && n > 1, k, If[n + m ==d, -1, If[n + m - 2 == d, -1, If[n == 1 && m == d, -k, 0]]]]
%e Triangular sequence:
%e {-1},
%e {-1, -1},
%e {2, 2, 1},
%e {1, 1, 1, -1},
%e {1, 0, -1, 2, 1},
%e {2, 0, 0, 4,1, -1},
%e {-1, 0, -1, -6, -4, 2, 1},
%e {1, 1, -1, -7, -3, 7, 1, -1},
%e {-2, -2, -1, 8, 6, -12, -7, 2, 1},
%e {-1, -1, -2, 10, 3, -23, -6, 10, 1, -1},
%e {-1, 0, 2, -12, -7, 34, 22, -18, -10, 2, 1}
%t k = 1; An[d_] := Table[If[n + m - 1 == d && n > 1, k, If[n + m == d, -1, If[n + m - 2 == d, -1, If[n == 1 &&m == d, -k, 0]]]], {n, 1, d}, {m, 1, d}]; Join[An[1], Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d,1, 20}]]; Flatten[%]
%K uned,sign,tabl
%O 1,4
%A _Roger L. Bagula_, Nov 02 2006
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