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A158800 A characteristic polynomial triangle of an Hadamard matrix self-similar lower triangular system: H(2^(n-1))->H(2^n) 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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}}

The system works orthogonally such that:

H(2^n].H(2^n]=IdentityMatrix[2^n]

Table[HadamardMatrix[2^n].HadamardMatrix[2^n], {n, 1, 4}]

The orthogonality allows one to form lower triangular quantum Heisenberg matrix mechanical Hamiltonian systems:

H(2^n)*Hamiltonian(2^n)*H(2^n)=E(2^n)*IdentityMatrix[2^n]

Hamiltonian(2^n)=Kinetic(2^n)+Potential(2^n).

LINKS

Table of n, a(n) for n=0..71.

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).

EXAMPLE

{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}

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[%]

CROSSREFS

Sequence in context: A308400 A236541 A113206 * A144024 A185249 A075107

Adjacent sequences:  A158797 A158798 A158799 * A158801 A158802 A158803

KEYWORD

sign,tabl,uned

AUTHOR

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

STATUS

approved

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Last modified September 21 19:40 EDT 2021. Contains 347598 sequences. (Running on oeis4.)