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.

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

Offset: 0

Author

Roger L. Bagula, Mar 19 2008

Keywords

• tabl
• sign

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).

Examples

			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;
		

Crossrefs

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

Programs

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']