A158234 Coefficients of powers of two Hadamard characteristic polynomials: M(n)=Hadamard[2^n] except for 12.

1, 0, 1, -2, 0, 1, -16, -8, 0, 2, 1, -4096, -3072, -1792, -448, 0, 56, 28, 6, 1, -2985984, -2488320, -1824768, -760320, -253440, -46464, 0, 3872, 1760, 440, 88, 10, 1, 4294967296, 0, 671088640, 0, 29360128, 0, -2752512, 0, -344064, 0, -10752, 0, 448

Offset

0,4

Comments

Row sums are:

{1, 1, -1, -21, -9317, -8353125, 4992309225,...}.

The program uses an add on :

Hadamard.m

downloaded at: http://ftp2.de.freebsd.org/pub/math/mathematica/math-source/Applications/Mathematics/Applied/0205-760/Hadamard.m

This method gives random solutions for higher Hadamard matrices

without the matrix self-similar qualification.

Example matrix: Hadamard[12][[1]]={{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},

{1, -1, 1, 1, 1, -1, 1, -1, -1, 1, -1, -1},

{1, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1},

{1, -1, 1, -1, -1, -1, -1, 1, -1, 1, 1, 1},

{1, -1, -1, 1, -1, 1, 1, -1, -1, -1, 1, 1},

{1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, 1},

{1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, -1},

{1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1},

{1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1},

{1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1},

{1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1},

{ 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, 1, -1}}

Links

Table of n, a(n) for n=0..45.

Formula

M(n)=Hadamard[2^n] with 12 added;

out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x).

Example

{1},
{0, 1},
{-2, 0, 1},
{-16, -8, 0, 2, 1},
{-4096, -3072, -1792, -448, 0, 56, 28, 6, 1},
{-2985984, -2488320, -1824768, -760320, -253440, -46464, 0, 3872, 1760, 440, 88, 10, 1},
{4294967296, 0, 671088640, 0, 29360128, 0, -2752512, 0, -344064, 0, -10752, 0, 448, 0, 40, 0, 1}

Mathematica

Needs["Hadamard`"];
Table[If[Hadamard[n] == {} && n >= 3, 0, If[n == 2, Hadamard[2], Hadamard[n][[1]]]], {n, 1, 10}];
a = Join[{1}, {x}, Union[ Table[CharacteristicPolynomial[If[Hadamard[n] == {} && n >= 3, 0, If[n == 2, Hadamard[2], Hadamard[n][[1]]]], x], {n, 2, 16}]]];
Table[CoefficientList[a[[n]], x], {n, 1, Length[a] - 1}];
Flatten[%]
Table[Apply[Plus, CoefficientList[a[[n]], x]], {n, 1, Length[a] - 1}];

Crossrefs

Sequence in context: A328924 A217654 A267164 * A270882 A278778 A356817

Adjacent sequences: A158231 A158232 A158233 * A158235 A158236 A158237

Keyword

sign,tabl,uned,obsc

Author

Roger L. Bagula, Mar 14 2009

Status

approved