A158566 A complex matrix self-similar coefficient set of the imaginary part based on the Hadamard matrix pattern: {{1,1},{1,I}}.

1, 0, -1, 1, 0, 2, -1, -2, 1, 0, 16, -32, -24, 36, 12, -12, -2, 1, 0, 0, 15360, -61440, 64256, 30720, -75456, 3328, 33552, -4608, -7776, 960, 984, -64, -60, 0, 1, 0, 0, 0, 0, 0, 0, -738734374912, 3272765079552, -5038533509120, 1561623265280

Offset

0,6

Comments

Row sums are:

{1, 0, 0, -5, -243, -27275755, 120788025582872005936447545,...}.

Example Matrix:

M(2^2)={{0, 0, 0, 0},

{0, 1, 0, 1},

{0, 0, 1, 1},

{0, 1, 1, 0}}.

The real part resembles the Hadamard {0,1} types and the

imaginary part resembles the Cantor-Hadamard difference set.

Links

Table of n, a(n) for n=0..44.

Formula

M(2)={{1,1},

{1,I}}

M(2)->{{M(2),M(2)},

{M(2),I*M(2)}}

out_(n,m)=coefficients(characteristicpolynomial(M(2*n),x),x)

Example

{1},
{0, -1, 1},
{ 0, 2, -1, -2, 1},
{0, 16, -32, -24, 36, 12, -12, -2, 1},
{0, 0, 15360, -61440, 64256, 30720, -75456, 3328, 33552, -4608, -7776, 960, 984, -64, -60, 0, 1},
{0, 0, 0, 0, 0, 0, -738734374912, 3272765079552, -5038533509120, 1561623265280, 4173521223680, -4536982831104, 243127025664, 1959101726720, -804463575040, -341990440960, 273888903168, 17242980352, -48180428800, 3289825280, 5312655360, -697614336, -396166144, 62059520, 20684800, -3114240, -755392, 88192, 18160, -1200, -240, 4, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -84058123395079684803788800000, 325607168993862037484339200000, -354316205730657686855352320000, -190162614715436308287717376000, 655456777374702426148936089600, -279039730465071392233812393984, -364172687113877637262106165248, 370669303608131315888746921984, 32208515725219400365310476288, -182231402145693428780892160000, 52083018181929181499810119680, 44041293190905969729727365120, -27702188053015447532323471360, -3727533719505386024688680960, 6930033499184341158996213760, -792092738592128283378188288, -1010984680306355608261492736, 299536460359127616295272448, 84786557096026665937534976, -47806122844238883953049600, -2485272478725396152451072, 4794912149374421356773376, -318569650484391914242048, -330827172552143552905216, 51310439278712350310400, 16068630084787200065536, -3982300845137658380288, -536192426749120741376, 207822462939119484928, 10765438286697594880,
-7924215175834501120, -32150582077685760, 228132520724398080, -5906159815884800, -5029151301959680, 223665666129920, 85210462945280, -4620068454400, -1103983820800, 61876674560, 10754044928, -543453184, -75890688, 2950144, 359680, -8448, -976, 8, 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, 1}, {1, I}};
HadamardMatrix[n_] := Module[{m}, m = {{1, 1}, {1, I}}; KroneckerProduct[m, HadamardMatrix[n/2]]];
Table[Im[HadamardMatrix[2^n]], {n, 1, 4}];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ Im[HadamardMatrix[2^n]], x], x], {n, 1, 6}]];
Flatten[%]
Join[{1}, Table[Apply[Plus, CoefficientList[CharacteristicPolynomial[Im[ HadamardMatrix[2^n]], x], x]], {n, 1, 6}]];

Crossrefs

Sequence in context: A269942 A094645 A105793 * A128410 A059782 A093654

Adjacent sequences: A158563 A158564 A158565 * A158567 A158568 A158569

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Mar 21 2009

Status

approved