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A158336 A triangle of matrix polynomials: m(n)=antisymmeticmatix(n).pseudotranspose[antisymmeticmatix(n)]. 0
1, 0, -1, -1, 0, 1, 0, 9, 0, -1, 1, 0, -34, 0, 1, 0, -25, 0, 90, 0, -1, -1, 0, 195, 0, -195, 0, 1, 0, 49, 0, -931, 0, 371, 0, -1, 1, 0, -644, 0, 3334, 0, -644, 0, 1, 0, -81, 0, 4788, 0, -9846, 0, 1044, 0, -1, -1, 0, 1605, 0, -25290, 0, 25290, 0, -1605, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

The pseusotranspose operation used is: pseudotranspose[a(n)]=Reverse[I(n)].a(n).

Row sums are:

{1, -1, 0, 8, -32, 64, 0, -512, 2048, -4096, 0,...}. Unsigned row sums are:

{1, 1, 2, 10, 36, 116, 392, 1352, 4624, 15760, 53792,...}.

Example matrix is:

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

{-1, 2, -1},

{2, -1, -1}}

LINKS

Table of n, a(n) for n=0..65.

FORMULA

m(n)=antisymmeticmatix(n).pseudotranspose[antisymmeticmatix(n)].;

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

EXAMPLE

{1},

{0, -1},

{-1, 0, 1},

{0, 9, 0, -1},

{1, 0, -34, 0, 1},

{0, -25, 0, 90, 0, -1},

{-1, 0, 195, 0, -195, 0, 1},

{0, 49, 0, -931, 0, 371, 0, -1},

{1, 0, -644, 0, 3334, 0, -644, 0, 1},

{0, -81, 0, 4788, 0, -9846, 0, 1044, 0, -1},

{-1, 0, 1605, 0, -25290, 0, 25290, 0, -1605, 0, 1}

MATHEMATICA

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

pt[a_] := Reverse[IdentityMatrix[Length[a]]].a;

T[n_, m_, d_] := If[ m < n, (-1)^(n + m), If[m > n, -(-1)^(n + m), 0]];

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

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

Table[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

A119467

Sequence in context: A221804 A275107 A274186 * A021530 A272232 A110909

Adjacent sequences:  A158333 A158334 A158335 * A158337 A158338 A158339

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Mar 16 2009

STATUS

approved

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Last modified April 7 10:46 EDT 2020. Contains 333301 sequences. (Running on oeis4.)