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%I #6 Mar 05 2014 03:23:19
%S 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,
%T 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,
%U -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
%N 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).
%C It was a real problem getting the matrices to agree with the polynomials: I was getting shift function polynomials!)
%C 1 X 1 {{1}}
%C 2 X 2 {{0, 1}, {1, -1}}
%C 3 X 3 {{0, 1, 0}, {0, 0, 1}, {1, -1, 0}}
%C 4 X 4 {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, -1, -1, 2}}
%C 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}}
%F 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)
%e Input triangular sequence from the recurvise polynomials:
%e {{1},
%e {-1,1},
%e {-1, 1, -1},
%e {-1, 1, 0, -1},
%e {-1, 1, 1, -2, 1},
%e {-1, 1, 2, -3, 1, 1}
%e Output triangular sequence from characteristic polynomials of matrices:
%e {1},
%e {1, -1},
%e {-1, 1, 1},
%e {1, -1, 0, -1},
%e {-1, 1, 1, -2, 1},
%e {1, -1, -2, 3, -1, -1}
%t 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[%]
%K tabl,uned,sign
%O 1,14
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 24 2006
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