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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 (list; table; graph; refs; listen; history; text; internal format)
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).

LINKS

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

FORMULA

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

upper sub-diagonal: 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 sub-diagonal: 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

Adjacent sequences:  A135666 A135667 A135668 * A135670 A135671 A135672

KEYWORD

sign,tabl,less

AUTHOR

Roger L. Bagula, Feb 16 2008

EXTENSIONS

Edited by G. C. Greubel, May 23 2019

STATUS

approved

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Last modified May 16 02:41 EDT 2021. Contains 343937 sequences. (Running on oeis4.)