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A135669 Triangular sequence of coefficients of characteristic polynomials of a tridiagonal matrix.

%I A135669 #22 Mar 31 2024 17:28:52
%S A135669 -1,1,-1,1,-1,1,1,-3,3,-1,1,-5,8,-5,1,1,-7,16,-16,7,-1,1,-9,27,-38,27,
%T A135669 -9,1,1,-11,41,-75,75,-41,11,-1,1,-13,58,-131,170,-131,58,-13,1,1,-15,
%U A135669 78,-210,336,-336,210,-78,15,-1,1,-17,101,-316,602,-742,602,-316,101,-17,1
%N A135669 Triangular sequence of coefficients of characteristic polynomials of a tridiagonal matrix.
%C A135669 The first few characteristic polynomials associated with this matrix are:
%C A135669    1 - x,
%C A135669    1 - x + x^2,
%C A135669   (1 - x)^3,
%C A135669   (1 - x)^2*(1 - 3*x + x^2),
%C A135669   (1 - x)^3*(1 - 4*x + x^2),
%C A135669   (1 - x)^4*(1 - 5*x + x^2).
%H A135669 G. C. Greubel, <a href="/A135669/b135669.txt">Rows n = 1..100 of triangle, flattened</a>
%F A135669 With the sequence function -(n+1) the tridiagonal matrix is formed by
%F A135669 upper subdiagonal: c(n,k) = if(n=1, -(k+1), 0),
%F A135669 diagonal: a(n,k) = if(n=1 & k > 1, -1, if(n=2, k, 1)),
%F A135669 lower subdiagonal: b(n) = 1,
%F A135669 where the triangle is formed by t(k) = M(i, j, k) for 1 <= j <= k, 1 <= i <= k,
%F A135669 and T(n) = coefficients of CharacteristicPolynomial(M(n), x).
%e A135669 Triangle begins:
%e A135669   -1;
%e A135669    1,  -1;
%e A135669    1,  -1,   1;
%e A135669    1,  -3,   3,   -1;
%e A135669    1,  -5,   8,   -5,   1;
%e A135669    1,  -7,  16,  -16,   7,   -1;
%e A135669    1,  -9,  27,  -38,  27,   -9,   1;
%e A135669    1, -11,  41,  -75,  75,  -41,  11,   -1;
%e A135669    1, -13,  58, -131, 170, -131,  58,  -13,   1;
%e A135669    1, -15,  78, -210, 336, -336, 210,  -78,  15,  -1;
%e A135669    1, -17, 101, -316, 602, -742, 602, -316, 101, -17, 1;
%t A135669 a[n_, k_]:= If[n==1 && k > 1, -1, If[n==2, k, 1]];
%t A135669 c[n_, k_]:= If[n==1, -k-1, 0];
%t A135669 T[n_, m_, k_]:= If[n==m, a[n,k], If[n==m-1,1, If[n==m+1, c[n-1,k], 0]]];
%t A135669 M0[k_]:= Table[T[n, m, k], {n, 1, k}, {m, 1, k}];
%t A135669 TableForm[Table[M0[n], {n, 1, 4}]];
%t A135669 TableForm[Table[Inverse[M0[n]], {n, 1, 4}]];
%t A135669 Table[Factor[CharacteristicPolynomial[M0[n], x]], {n, 1, 10}];
%t A135669 Join[{{-1}}, Table[CoefficientList[CharacteristicPolynomial[M0[n], x], x], {n, 1, 10}]]//Flatten (* modified by _G. C. Greubel_, May 23 2019 *)
%K A135669 sign,tabl,less
%O A135669 1,8
%A A135669 _Roger L. Bagula_, Feb 16 2008
%E A135669 Edited by _G. C. Greubel_, May 23 2019