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A139584 A triangle of coefficients of A053122 type binomials {x,y},{y,z} and {z,x}, made using A_n Cartan type matrix characteristic polynomials: an(x,n) = CharacteristicPolynomial(M(A_n,n)); f(x,y,n) = Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}]; p(x,y,z,n) = f(x,y,n) + f(y,z,n) + f(z,x,n).

%I A139584 #8 Dec 09 2017 19:43:21
%S A139584 3,5,-2,6,-8,2,7,-20,12,-2,9,-40,42,-16,2,12,-70,112,-72,20,-2,15,
%T A139584 -112,252,-240,110,-24,2,17,-168,504,-660,440,-156,28,-2,18,-240,924,
%U A139584 -1584,1430,-728,210,-32,2,19,-330,1584,-3432,4004,-2730,1120,-272,36,-2,21,-440,2574,-6864,10010,-8736,4760,-1632,342
%N A139584 A triangle of coefficients of A053122 type binomials {x,y},{y,z} and {z,x}, made using A_n Cartan type matrix characteristic polynomials: an(x,n) = CharacteristicPolynomial(M(A_n,n)); f(x,y,n) = Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}]; p(x,y,z,n) = f(x,y,n) + f(y,z,n) + f(z,x,n).
%C A139584 Row sums are: {3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, ...}
%F A139584 an(x,n) = CharacteristicPolynomial(M(A_n,n));
%F A139584 f[x,y,n) = Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}];
%F A139584 p(x,y,z,n) = f(x,y,n) + f(y,z,n) + f(z,x,n);
%F A139584 Out_n,m = Coefficients(p(x,1,1,n).
%e A139584 {3},
%e A139584 {5, -2},
%e A139584 {6, -8, 2},
%e A139584 {7, -20,12, -2},
%e A139584 {9, -40, 42, -16, 2},
%e A139584 {12, -70, 112, -72, 20, -2},
%e A139584 {15, -112, 252, -240, 110, -24, 2},
%e A139584 {17, -168, 504, -660, 440, -156, 28, -2},
%e A139584 {18, -240, 924, -1584, 1430, -728, 210, -32, 2},
%e A139584 {19, -330, 1584, -3432, 4004, -2730, 1120, -272,36, -2},
%e A139584 {21, -440, 2574, -6864, 10010, -8736, 4760, -1632, 342, -40, 2}
%t A139584 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
%t A139584 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; an[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]]
%t A139584 f[x_, y_, n_] := Sum[CoefficientList[an[x, n], x][[i + 1]]*x^i*y^(n - i), {i, 0, Length[CoefficientList[an[x, n], x]] - 1}];
%t A139584 Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}];
%t A139584 a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]
%Y A139584 Cf. A053122.
%K A139584 uned,sign
%O A139584 1,1
%A A139584 _Roger L. Bagula_ and _Gary W. Adamson_, Jun 11 2008