A173814 Coefficients of Hadamard Cartan G_2 self-similar 2^n matrices:M={{2, -1}, {-3, 2}}.

1, 1, -4, 1, 1, -16, 30, -16, 1, 1, -64, 676, -2752, 4678, -2752, 676, -64, 1, 1, -256, 13560, -316160, 3830300, -25002240, 87841480, -180202240, 227671110, -180202240, 87841480, -25002240, 3830300, -316160, 13560, -256, 1, 1, -1024

Offset: 0

Author

Roger L. Bagula, Feb 25 2010

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

{1, -2, 0, 400, 0, 231040000000000, 0,...}.

Examples

			{1},
{1, -4, 1},
{1, -16, 30, -16, 1},
{1, -64, 676, -2752, 4678, -2752, 676, -64, 1},
{1, -256, 13560, -316160, 3830300, -25002240, 87841480, -180202240, 227671110, -180202240, 87841480, -25002240, 3830300, -316160, 13560, -256, 1},
{1, -1024, 255376, -30325760, 2060069240, -86239093760, 2306160223920, -40571580718080, 489632650203420, -4209374685189120, 26512089196724880, -124638699726597120, 442120325884773960, -1188638208146519040, 2420933452415430960, -3721572797083978752, 4298314898249481798, -3721572797083978752, 2420933452415430960, -1188638208146519040, 442120325884773960, -124638699726597120, 26512089196724880, -4209374685189120, 489632650203420, -40571580718080, 2306160223920, -86239093760, 2060069240, -30325760, 255376, -1024, 1}, ...
		

Crossrefs

Cf. A136674, A158800

Programs

Mathematica

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] := {{2, -1}, {-3, 2}}
HadamardMatrix[n_] := Module[{m},
m = {{2, -1}, {-3, 2}};
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

M(2)={{2, -1}, {-3, 2}};

M(4)={{4, -2, -2, 1}, {-6, 4, 3, -2}, {-6, 3, 4, -2}, {9, -6, -6, 4}},etc.