OFFSET
1,14
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}}
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)
EXAMPLE
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}
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[%]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 24 2006
STATUS
approved