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A158239 a triangle of coefficients Pseudo-Hadamard matrices as integer characteristic polynomials ( the code and initial values are very long, but the basic recurrence is the Hadamard matrix self-similarity). 0
1, 1, -1, -2, 0, 1, -4, 2, 2, -1, 16, 0, -8, 0, 1, 32, -16, -16, 8, 2, -1, -128, 0, 80, 0, -16, 0, 1, -64, 32, 64, -32, -20, 10, 2, -1, 4096, 0, -2048, 0, 384, 0, -32, 0, 1, 12288, -4096, -6144, 2048, 1152, -384, -96, 32, 3, -1, -32768, 0, 20480, 0, -5120, 0, 640, 0, -40, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums are: {1, 0, -1, -1, 9, 9, -63, -9, 2401, 4802, -16807, 33614, -8353125,...}.

These are defined to give an orthogonality between where the ordinary {1,-1} Hadamard matrices.

A better diagonal transpose version is gotten using in H(5) and H(11) Sqrt[2] for 2

and in H(9) Sqrt[8] for three. What I did was use blocks of matrices in the odd Hadamard matrices,

so that like U(1)*SU(2) they behave as if they are orthogonal.

Source : my code uses Chinese student's code as base :

http : // blade100.math.scu.edu.tw/~u9217/hw3.nb

My object here is to bridge the Sierpinski-Hadamard {0,1} like A122944

to the {1,-1} symbol version.

The code is long here:

the approach is more of a science and engineering one

than a pure mathematics one.

REFERENCES

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, pp. 44-48.

LINKS

Table of n, a(n) for n=0..64.

FORMULA

Basic recurrence for even values uses the matrix self-similarity of the Hadamard matrices.

EXAMPLE

{1},

{1, -1},

{-2, 0, 1},

{-4, 2, 2, -1},

{16, 0, -8, 0, 1},

{32, -16, -16, 8, 2, -1},

{-128, 0, 80, 0, -16, 0, 1},

{-64, 32, 64, -32, -20, 10, 2, -1},

{4096, 0, -2048, 0, 384, 0, -32, 0, 1},

{12288, -4096, -6144, 2048, 1152, -384, -96, 32, 3, -1},

{-32768, 0, 20480, 0, -5120, 0, 640, 0, -40, 0, 1},

{32768, 32768, -20480, -20480, 5120, 5120, -640, -640, 40, 40, -1, -1},

{-2985984, -2488320, -1824768, -760320, -253440, -46464, 0, 3872, 1760, 440, 88, 10, 1}

MATHEMATICA

(* source : uses Chinese student's code as base : http : // blade100.math.scu.edu.tw/~u9217/hw3.nb*)

Clear[HadamardMatrix, a];

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[0] := {{0}};

HadamardMatrix[1] := {{1}}

HadamardMatrix[2] := {{1, 1},

{1, -1}};

HadamardMatrix[ 3] := {{2, 0, 0},

{0, 1, 1},

{0, 1, -1}};

HadamardMatrix[5] := {{2, 0, 0, 0, 0},

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

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

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

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

HadamardMatrix[7] := {{1, 1, 0, 0, 0, 0, 0},

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

{0, 0, 2, 0, 0, 0, 0},

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

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

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

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

HadamardMatrix[9] := {{3, 0, 0, 0, 0, 0, 0, 0, 0},

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

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

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

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

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

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

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

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

HadamardMatrix[11] := {{2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1},

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

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

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

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

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

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

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

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

HadamardMatrix[12] := {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},

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

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

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

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

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

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

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

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

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

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

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

HadamardMatrix[n_] := Module[{m}, m = {{1, 1}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]];

Table[HadamardMatrix[n], {n, 0, 12}];

Table[HadamardMatrix[n].Transpose[HadamardMatrix[n]], {n, 0, 12}];

Table[CharacteristicPolynomial[HadamardMatrix[n], x], {n, 1, 12}];

a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[HadamardMatrix[n], x], x], {n, 1, 12}]];

Flatten[%]

Table[Apply[Plus, a[[n]]], {n, 1, Length[a]}];

CROSSREFS

A158234, A122944

Sequence in context: A194686 A266213 A124915 * A159819 A030188 A176703

Adjacent sequences:  A158236 A158237 A158238 * A158240 A158241 A158242

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Mar 14 2009

STATUS

approved

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Last modified June 26 15:16 EDT 2017. Contains 288766 sequences.