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

-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, -9, 1, 1, -11, 41, -75, 75, -41, 11, -1, 1, -13, 58, -131, 170, -131, 58, -13, 1, 1, -15, 78, -210, 336, -336, 210, -78, 15, -1, 1, -17, 101, -316, 602, -742, 602, -316, 101, -17, 1

Offset: 1

Author

Roger L. Bagula, Feb 16 2008

Keywords

• sign
• tabl
• less

Comments

The first few characteristic polynomials associated with this matrix are:

1 - x,

1 - x + x^2,

(1 - x)^3,

(1 - x)^2*(1 - 3*x + x^2),

(1 - x)^3*(1 - 4*x + x^2),

(1 - x)^4*(1 - 5*x + x^2).

Examples

			Triangle begins:
  -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,   -9,   1;
   1, -11,  41,  -75,  75,  -41,  11,   -1;
   1, -13,  58, -131, 170, -131,  58,  -13,   1;
   1, -15,  78, -210, 336, -336, 210,  -78,  15,  -1;
   1, -17, 101, -316, 602, -742, 602, -316, 101, -17, 1;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Programs

Mathematica

a[n_, k_]:= If[n==1 && k > 1, -1, If[n==2, k, 1]];
c[n_, k_]:= If[n==1, -k-1, 0];
T[n_, m_, k_]:= If[n==m, a[n,k], If[n==m-1,1, If[n==m+1, c[n-1,k], 0]]];
M0[k_]:= Table[T[n, m, k], {n, 1, k}, {m, 1, k}];
TableForm[Table[M0[n], {n, 1, 4}]];
TableForm[Table[Inverse[M0[n]], {n, 1, 4}]];
Table[Factor[CharacteristicPolynomial[M0[n], x]], {n, 1, 10}];
Join[{{-1}}, Table[CoefficientList[CharacteristicPolynomial[M0[n], x], x], {n, 1, 10}]]//Flatten (* modified by G. C. Greubel, May 23 2019 *)

Formula

With the sequence function -(n+1) the tridiagonal matrix is formed by

upper subdiagonal: c(n,k) = if(n=1, -(k+1), 0),

diagonal: a(n,k) = if(n=1 & k > 1, -1, if(n=2, k, 1)),

lower subdiagonal: b(n) = 1,

where the triangle is formed by t(k) = M(i, j, k) for 1 <= j <= k, 1 <= i <= k,

and T(n) = coefficients of CharacteristicPolynomial(M(n), x).

Extensions

Edited by G. C. Greubel, May 23 2019