A158335 A triangle of matrix polynomials: m(n)=antisymmeticmatix(n).Transpose[antisymmeticmatix(n)].

1, 0, -1, 1, -2, 1, 0, -9, 6, -1, 1, -12, 38, -12, 1, 0, -25, 100, -110, 20, -1, 1, -30, 255, -452, 255, -30, 1, 0, -49, 490, -1519, 1484, -511, 42, -1, 1, -56, 924, -3976, 6470, -3976, 924, -56, 1, 0, -81, 1512, -9324, 21816, -21942, 9240, -1548, 72, -1, 1, -90

Offset: 0

Author

Roger L. Bagula, Mar 16 2009

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

{1, -1, 0, -4, 16, -16, 0, -64, 256, -256, 0,...}. Unsigned row sums are:

{1, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144,...}.

Example matrix is:

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

{-1, 2, -1},

{-1, -1, 2}}

Examples

			{1},
{0, -1},
{1, -2, 1},
{0, -9, 6, -1},
{1, -12, 38, -12, 1},
{0, -25, 100, -110, 20, -1},
{1, -30, 255, -452, 255, -30, 1},
{0, -49, 490, -1519, 1484, -511, 42, -1},
{1, -56, 924, -3976, 6470, -3976, 924, -56, 1},
{0, -81, 1512, -9324, 21816, -21942, 9240, -1548, 72, -1},
{1, -90, 2445, -19320, 63090, -92252, 63090, -19320, 2445, -90, 1}
		

Crossrefs

A119467

Programs

Mathematica

Clear[M, T, d, a, x, a0];
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}].Transpose[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).Transpose[antisymmeticmatix(n)];

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