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
Examples
{1},
{2, -4, 1},
{ 16, -64, 56, -16, 1},
{4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 1}, ...
Programs
-
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[%]
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.
Comments