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Binomial centered tridigonal matrices as a triangular sequence: t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]].
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%I #6 Mar 12 2014 16:37:03

%S 1,1,-1,0,-2,1,0,-3,4,-1,3,-14,19,-8,1,48,-173,204,-89,16,-1,1505,

%T -4866,5173,-2082,381,-32,1,108780,-325990,316978,-113481,18926,-1580,

%U 64,-1,19072536,-53887686,48428411,-15201276,2206536,-164222,6469,-128,1,8332293760,-22465873081,18859204368,-5176293234

%N Binomial centered tridigonal matrices as a triangular sequence: t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]].

%C These are pretty matrices in terms of symmetry. Matrices: 1 X 1 {{1}} 2 X 2 {{1, -1}, {-1, 1}} 3 X 3 {{1, -1, 0}, {-1, 2, -1}, {0, -1, 1}} 4 X 4 {{1, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 1}} 5 X 5 {{1, -1, 0, 0, 0}, {-1, 4, -1, 0, 0}, {0, -1, 6, -1, 0}, {0, 0, -1, 4, -1}, {0, 0, 0, -1, 1}} 6 X 6 {{1, -1, 0, 0, 0, 0}, {-1, 5, -1, 0, 0, 0}, {0, -1, 10, -1, 0, 0}, {0, 0, -1, 10, -1, 0}, {0, 0, 0, -1, 5, -1}, {0, 0, 0, 0, -1, 1}}

%F t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]]

%e Triangular sequence:

%e {1},

%e {1, -1},

%e {0, 2, 1},

%e {0, 3, 2, -1},

%e {3, -4, -11, 2, 1},

%e {48, -13, -106, 21, 6, -1},

%e {-1505, 36, 2693, -58, -129, 2, 1},

%e {-108780, 5530, 171342, -8705, -5290, 268,20, -1}

%t An[d_] := Table[If[n + m - 1 == d, Binomial[d - 1, n - 1], If[n + m ==d, -1, If[n + m - 2 == d, -1, 0]]], {n, 1, d}, {m, 1, d}]; Join[An[1], Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

%K uned,sign

%O 1,5

%A _Roger L. Bagula_, Nov 01 2006