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A136451 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n matrix: 2 on the main antidiagonal, -1 on the adjacent sub-antidiagonals and 0 otherwise. 2
1, 2, -1, -3, 2, 1, -4, 6, 2, -1, 5, -10, -9, 2, 1, 6, -19, -16, 12, 2, -1, -7, 28, 42, -22, -15, 2, 1, -8, 44, 68, -74, -28, 18, 2, -1, 9, -60, -138, 126, 115, -34, -21, 2, 1, 10, -85, -208, 316, 202, -165, -40, 24, 2, -1, -11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

We start from tri-antidiagonal variants of the Cartan A-n group matrix. For n=1 this is {2}, for n=2 this is {{-1,2},{2,-1}}, for n=3 {{0,-1,2},{-1,2,-1},{2,-1,0}}, for n =4 {{0,0,-1,2},{0,-1,2,-1},{-1,2,-1,0},{2,-1,0,0}} etc. The n-th row of the triangle are the expansion coefficients of the characteristic polynomial.

For n=0, the empty product of the empty matrix is assigned the value T(0,0)=1.

Row sums (characteristic polynomials evaluated at x=0) are 1, 1, 0, 3, -11, -16, 29, 21, 0, 55, -199, -288, 521, 377, 0, 987, -3571, -5168, 9349, 6765, 0, ... (see A038150).

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

1;

2, -1;

-3,2, 1;

-4, 6, 2, -1;

5, -10, -9, 2, 1;

6, -19, -16, 12, 2, -1;

-7,28, 42, -22, -15, 2, 1;

-8, 44, 68, -74, -28,18, 2, -1;

9, -60, -138, 126, 115, -34, -21, 2, 1;

10, -85, -208,316, 202, -165, -40, 24, 2, -1;

-11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1;

MAPLE

A136451x := proc(n, x)

    local A, r, c ;

    A := Matrix(1..n, 1..n) ;

    for r from 1 to n do

    for c from 1 to n do

            A[r, c] :=0 ;

        if r+c = 1+n then

            A[r, c] := A[r, c]+2 ;

        elif abs(r+c-1-n)= 1 then

            A[r, c] :=  A[r, c]-1 ;

        end if;

    end do:

    end do:

    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A, x) ;

end proc;

A136451 := proc(n, k)

    coeftayl( A136451x(n, x), x=0, k) ;

end proc:

seq(seq(A136451(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Dec 04 2011

MATHEMATICA

H[n_] := Table[Table[If[i + j - 1 == n, 2, If[i + j - 1 == n + 1, -1, If[i + j - 1 == n - 1, -1, 0]]], {i, 1, n}], {j, 1, n}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[H[n], x], x], {n, 1, 10}]]; Flatten[a']

CROSSREFS

Cf. A124018 (variant), A005993 (column k=1), A061927 (bisection column k=2).

Sequence in context: A208825 A344391 A089353 * A066121 A039911 A208945

Adjacent sequences:  A136448 A136449 A136450 * A136452 A136453 A136454

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Mar 19 2008

STATUS

approved

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Last modified June 20 21:35 EDT 2021. Contains 345255 sequences. (Running on oeis4.)