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

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, 0, 28, 0, -8, 0, 1, 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, 1, 0, -32, 0, 1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Mar 27 2009

Keywords

• sign
• tabf
• uned

Comments

Row sums are zero except for n=0.

Example matrix:

H(8)={{1, 0, 0, 0, 0, 0, 0, 0},

{1, -1, 0, 0, 0, 0, 0, 0},

{1, 0, -1, 0, 0, 0, 0, 0},

{1, -1, -1, 1, 0, 0, 0, 0},

{1, 0, 0, 0, -1, 0, 0, 0},

{1, -1, 0, 0, -1, 1, 0, 0},

{1, 0, -1, 0, -1, 0, 1, 0},

{1, -1, -1, 1, -1, 1, 1, -1}}

H[2^n] * H[2^n] = IdentityMatrix[2^n]

Examples

			{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, 0, 28, 0, -8, 0, 1},
{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},
{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}
		

Programs

Mathematica

Clear[HadamardMatrix];
MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]];
KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2},
M1 = M;
N1 = N;
LM = Length[M1];
LN = Length[N1];
Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}];
Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}];
N2 = {};
Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}];
N2 = Flatten[N2];
Partition[N2, LM*LN, LM*LN]]
HadamardMatrix[2] := {{1, 0}, {1, -1}};
HadamardMatrix[n_] := Module[{m}, m = {{1, 0}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]];
Table[HadamardMatrix[2^n], {n, 1, 4}];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ HadamardMatrix[2^n], x], x], {n, 1, 6}]];
Flatten[%]

Formula

Pattern Matrix:

H(2) = {{1, 0},

{1, -1}}

Iteration Matrix:

m = {{1, 0},

{1, -1}}

Matrix_Self_similar_Operator[H[2^(n-1)] = H(2^n).