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A158285 A triangle of coefficients of matrix polynomials:M(d)=If[ m == n, -1, 1] (odd orthogonality) 0
1, -1, -1, 0, 2, 1, 4, 0, -3, -1, -16, -16, 0, 4, 1, 48, 80, 40, 0, -5, -1, -128, -288, -240, -80, 0, 6, 1, 320, 896, 1008, 560, 140, 0, -7, -1, -768, -2560, -3584, -2688, -1120, -224, 0, 8, 1, 1792, 6912, 11520, 10752, 6048, 2016, 336, 0, -9, -1, -4096, -17920 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, -2, 3, 0, -27, 162, -729, 2916, -10935, 39366, -137781,...}.

These matrices are an all ones matrix with central diagonal negative.

These matrices have the odd orthogonality property of:

M(n).M(n)-M(n)-(-5+n)*Ones(n)=6*IdentityMatirix[n];

Table[T[n, m, d], {n, 1, d}, {m, 1, d}].Table[T[n, m, d], {n, 1, d}, {m, 1, d}]

- Table[T[n, m, d], {n, 1, d}, {m, 1, d}]

- (-5 + d)*Table[1, {n, d}, {m, d}]

Matrix example:

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

{1, -1, 1},

{1, 1, -1}}

LINKS

Table of n, a(n) for n=0..56.

FORMULA

M(d)=If[ m == n, -1, 1];

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

EXAMPLE

{1},

{-1, -1},

{0, 2, 1},

{4, 0, -3, -1},

{-16, -16, 0, 4, 1},

{48, 80, 40, 0, -5, -1},

{-128, -288, -240, -80, 0, 6, 1},

{320, 896, 1008, 560, 140, 0, -7, -1},

{-768, -2560, -3584, -2688, -1120, -224, 0, 8, 1},

{1792, 6912, 11520, 10752, 6048, 2016, 336, 0, -9, -1},

{-4096, -17920, -34560, -38400, -26880, -12096, -3360, -480, 0, 10, 1}

MATHEMATICA

Clear[M, T, d, a, x, a0];

T[n_, m_, d_] := If[ m == n, -1, 1];

M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];

a0 = Table[M[d], {d, 1, 10}];

Table[Det[M[d]], {d, 1, 10}];

Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];

a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];

Flatten[a]

Join[{1}, Table[Apply[Plus, CoefficientList[Expand[CharacteristicPolynomial[M[ n], x]], x]], {n, 1, 10}]];

CROSSREFS

Sequence in context: A048614 A001442 A226952 * A277994 A334112 A286238

Adjacent sequences:  A158282 A158283 A158284 * A158286 A158287 A158288

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Mar 15 2009

STATUS

approved

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Last modified October 1 08:39 EDT 2020. Contains 337442 sequences. (Running on oeis4.)