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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). 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, 1 (list; 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}}
H[2^n] * H[2^n] = IdentityMatrix[2^n]
LINKS
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: A369816 A236541 A113206 * A144024 A185249 A075107
KEYWORD
sign,tabf,uned
AUTHOR
STATUS
approved

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Last modified June 17 06:18 EDT 2024. Contains 373432 sequences. (Running on oeis4.)