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A158800 A characteristic polynomial triangle of a Hadamard matrix self-similar lower triangular system: H(2^(n-1)) -> H(2^n).

%I A158800 #15 Jan 30 2023 07:40:53
%S A158800 1,-1,0,1,1,0,-2,0,1,1,0,-4,0,6,0,-4,0,1,1,0,-8,0,28,0,-56,0,70,0,-56,
%T A158800 0,28,0,-8,0,1,1,0,-16,0,120,0,-560,0,1820,0,-4368,0,8008,0,-11440,0,
%U A158800 12870,0,-11440,0,8008,0,-4368,0,1820,0,-560,0,120,0,-16,0,1,1,0,-32,0,1
%N A158800 A characteristic polynomial triangle of a Hadamard matrix self-similar lower triangular system: H(2^(n-1)) -> H(2^n).
%C A158800 Row sums are zero except for n=0.
%C A158800 Example matrix:
%C A158800   H(8)={{1,  0,  0,  0,  0,  0,  0,  0},
%C A158800         {1, -1,  0,  0,  0,  0,  0,  0},
%C A158800         {1,  0, -1,  0,  0,  0,  0,  0},
%C A158800         {1, -1, -1,  1,  0,  0,  0,  0},
%C A158800         {1,  0,  0,  0, -1,  0,  0,  0},
%C A158800         {1, -1,  0,  0, -1,  1,  0,  0},
%C A158800         {1,  0, -1,  0, -1,  0,  1,  0},
%C A158800         {1, -1, -1,  1, -1,  1,  1, -1}}
%C A158800 H[2^n] * H[2^n] = IdentityMatrix[2^n]
%F A158800 Pattern Matrix:
%F A158800   H(2) = {{1,  0},
%F A158800           {1, -1}}
%F A158800 Iteration Matrix:
%F A158800      m = {{1,  0},
%F A158800           {1, -1}}
%F A158800 Matrix_Self_similar_Operator[H[2^(n-1)] = H(2^n).
%e A158800 {1},
%e A158800 {-1, 0, 1},
%e A158800 {1, 0, -2, 0, 1},
%e A158800 {1, 0, -4, 0, 6, 0, -4, 0, 1},
%e A158800 {1, 0, -8, 0, 28, 0, -56, 0, 70, 0, -56, 0, 28, 0, -8, 0, 1},
%e A158800 {1, 0, -16, 0, 120, 0, -560, 0, 1820, 0, -4368, 0, 8008, 0, -11440, 0, 12870, 0, -11440, 0, 8008, 0, -4368, 0, 1820, 0, -560, 0, 120, 0, -16, 0, 1},
%e A158800 {1, 0, -32, 0, 496, 0, -4960, 0, 35960, 0, -201376, 0, 906192, 0, -3365856, 0, 10518300, 0, -28048800, 0, 64512240, 0, -129024480, 0, 225792840, 0, -347373600, 0, 471435600, 0, -565722720, 0, 601080390, 0, -565722720, 0, 471435600, 0, -347373600, 0, 225792840, 0, -129024480, 0, 64512240, 0, -28048800, 0, 10518300, 0, -3365856, 0, 906192, 0, -201376, 0, 35960, 0, -4960, 0, 496, 0, -32, 0, 1}
%t A158800 Clear[HadamardMatrix];
%t A158800 MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]];
%t A158800 KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2},
%t A158800 M1 = M;
%t A158800 N1 = N;
%t A158800 LM = Length[M1];
%t A158800 LN = Length[N1];
%t A158800 Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}];
%t A158800 Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}];
%t A158800 N2 = {};
%t A158800 Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}];
%t A158800 N2 = Flatten[N2];
%t A158800 Partition[N2, LM*LN, LM*LN]]
%t A158800 HadamardMatrix[2] := {{1, 0}, {1, -1}};
%t A158800 HadamardMatrix[n_] := Module[{m}, m = {{1, 0}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]];
%t A158800 Table[HadamardMatrix[2^n], {n, 1, 4}];
%t A158800 Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ HadamardMatrix[2^n], x], x], {n, 1, 6}]];
%t A158800 Flatten[%]
%K A158800 sign,tabf,uned
%O A158800 0,7
%A A158800 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 27 2009