A173820 Coefficients of characteristic polynomials of Hadamard Cartan F_2 self-similar 2^n matrices:M={{2, -1}, {-2, 2}}.

1, 2, -4, 1, 16, -64, 56, -16, 1, 4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 1, 4294967296, -68719476736, 375809638400, -1043677052928, 1696981843968, -1726845288448, 1143073669120, -506453819392, 152912134144, -31653363712

Offset

0,2

Comments

Row sums are:

{1, -1, -7, -31, -208289151, 199276356275696712709633,

-27294457550222463310332530871924308277403810665846783,...}.

Links

Table of n, a(n) for n=0..27.

Formula

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

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

Example

{1},
{2, -4, 1},
{ 16, -64, 56, -16, 1},
{4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 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] := {{2, -1}, {-2, 2}}
HadamardMatrix[n_] := Module[{m},
m = {{2, -1}, {-2, 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[%]

Crossrefs

Cf. A136674, A158800, A173814, A136678

Sequence in context: A081538 A224783 A214058 * A030043 A045497 A117258

Adjacent sequences: A173817 A173818 A173819 * A173821 A173822 A173823

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Feb 25 2010

Status

approved