A123951 A polynomial of matrices is used to make a triangular sequence. The upper triangular antidiagonal Steinbach matrices are summed over their characteristic polynomial triangular sequences to give a new sequence of matrices: the characteristic polynomials of these new summed matrices are, then, used to make up this triangular sequence.

1, 1, -1, -1, -1, 1, -1, -3, 4, -1, 37, -88, 69, -19, 1, 10879, -14344, 6831, -1375, 99, -1, -4322473, -40529664, -17486038, 3188841, -40896, -2346, 1, -11384127259974047, -783824545942228, 1058675233347, 505084925760, -64007100, -32568519, 23164, -1, -121986767767877481129923

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 26 2006

Keywords

• uned
• tabl
• sign

Comments

Basically everything is done twice. The determinants get very large very fast for these matrices: Table[Det[w[[d]]], {d, 1, Length[w]}] {1, -1, -1, 37, 10879, -4322473, -11384127259974047, -121986767767877481129923, -323621163456130064854374309178100414058036559, 189651898964129252384795657180434913387386019400002936829101989683}

Examples

			{1},
{1, -1},
{-1, -1, 1},
{-1, -3, 4, -1},
{37, -88, 69, -19,1},
{10879, -14344, 6831, -1375, 99, -1},
{-4322473, -40529664, -17486038, 3188841, -40896, -2346, 1}
		

Links

• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31

Crossrefs

Cf. A122765, A122771.

Programs

Mathematica

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d],x], x], {d, 1, 20}]]; w = Join[{{{1}}}, Table[Sum[MatrixPower[a[[n]][[m + 1]]*An[n], m - 1], {m, 0, Length[a[[n]]] - 1}], {n, 2, 10}]]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[w[[d]], x], x], {d, 1, Length[w]}]]; Flatten[%]

Formula

p(n,x) = CharacteristicPolynomial(a(i,j)) p(n,x)->t(n,m) b(i,j) = Sum[t(i,j).a(j,k).{j,1,m}] p'(n,x) = CharacteristicPolynomial(b(i,j)) p'(n,x)->t'(n,m).