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A135669
Triangular sequence of coefficients of characteristic polynomials of a tridiagonal matrix.
5
-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
1,8
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).
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).
EXAMPLE
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;
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 *)
CROSSREFS
Sequence in context: A187371 A056611 A177407 * A132729 A196493 A251634
KEYWORD
sign,tabl,less
AUTHOR
Roger L. Bagula, Feb 16 2008
EXTENSIONS
Edited by G. C. Greubel, May 23 2019
STATUS
approved