A158336 A triangle of matrix polynomials: m(n)=antisymmeticmatix(n).pseudotranspose[antisymmeticmatix(n)].

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

Offset: 0

Author

Roger L. Bagula, Mar 16 2009

Keywords

• sign
• tabl
• uned

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}}

Examples

			{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}
		

Crossrefs

A119467

Programs

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}]];

Formula

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

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