A122945 Recursive polynomials (p(k, x) = p(k - 1, x) - x^2*p(k - 2, x) ) used to produce a set of matrices a(i,j) at level n that then produce the characteristic polynomials which provide the triangular sequence t(n,m).

1, 1, -1, -1, 1, 1, 1, -1, 0, -1, -1, 1, 1, -2, 1, 1, -1, -2, 3, -1, -1, -1, 1, 3, -4, 0, 3, 1, 1, -1, -4, 5, 2, -6, 2, -1, -1, 1, 5, -6, -5, 10, -2, -4, 1, 1, -1, -6, 7, 9, -15, 0, 10, -3, -1, -1, 1, 7, -8, -14, 21, 5, -20, 5, 5, 1, 1, -1, -8, 9, 20, -28, -14, 35, -5, -15, 4, -1, -1, 1, 9, -10, -27, 36, 28, -56, 0, 35, -9, -6, 1, 1, -1, -10, 11

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 24 2006

Keywords

• tabl
• uned
• sign

Comments

It was a real problem getting the matrices to agree with the polynomials: I was getting shift function polynomials!)

1 X 1 {{1}}

2 X 2 {{0, 1}, {1, -1}}

3 X 3 {{0, 1, 0}, {0, 0, 1}, {1, -1, 0}}

4 X 4 {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, -1, -1, 2}}

5 X 5 {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, -1, -2, 3, -1}}

Examples

			Input triangular sequence from the recurvise polynomials:
{{1},
{-1,1},
{-1, 1, -1},
{-1, 1, 0, -1},
{-1, 1, 1, -2, 1},
{-1, 1, 2, -3, 1, 1}
Output triangular sequence from characteristic polynomials of matrices:
{1},
{1, -1},
{-1, 1, 1},
{1, -1, 0, -1},
{-1, 1, 1, -2, 1},
{1, -1, -2, 3, -1, -1}
		

Programs

Mathematica

p[0, x] = 1; p[1, x] = x - 1; p[k_, x_] := p[k, x] = p[k - 1, x] - x^2*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 20}] ; An[d_] := Table[If[n == d, -w[[n]][[m]], If[m == n, 1, 0]], {n, 2, d}, {m, 1, d - 1}]; Table[An[d], {d, 2, 19}] b = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], \ {d, 2, 19}]]; Flatten[%]

Formula

p(k, x) = p(k - 1, x) - x^2*p(k - 2, x) p(k,n)->t0(i,j) t0(i,j)->a[i,j) a(i,j)->p'(n,x) p'(n,k)->t(n,m)