A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Nov 01 2006

Keywords

• tabl
• sign

Comments

The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.

Riordan array (1/(1-4*x+x^2), -x/(1-4*x+x^2)). - Philippe Deléham, Mar 04 2016

Examples

			Triangle begins as:
      1;
      4,     -1;
     15,     -8,     1;
     56,    -46,    12,    -1;
    209,   -232,    93,   -16,    1;
    780,  -1091,   592,  -156,   20,   -1;
   2911,  -4912,  3366, -1200,  235,  -24,  1;
  10864, -21468, 17784, -8010, 2120, -330, 28, -1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Joanne Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.

Crossrefs

Cf. A000302, A001353, A001906, A004254, A007070, A123966.

Cf. A139278, A159764, A181983, A207823 (absolute values).

Programs

Magma

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A124029:= func< n,k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
[A124029(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

Maple

A123966x := 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 then
            A[r,c] := A[r,c]+4 ;
        elif abs(r-c)= 1 then
            A[r,c] :=  A[r,c]-1 ;
        end if;
    end do:
    end do:
    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
end proc;
A123966 := proc(n,k)
    coeftayl( A123966x(n,x),x=0,k) ;
end proc:
seq(seq(A123966(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

Mathematica

(* Matrix version*)
k = 4;
T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Table[M[d], {d,10}]
Table[Det[M[d]], {d,10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}]
Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
(* Recursive Polynomial form*)
p[0, x]= 1; p[1, x]= (4-x); p[k_, x_]:= p[k, x]= (4-x)*p[k-1, x] - p[k -2, x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Additional program *)
Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

SageMath

def A124029(n,k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
flatten([[A124029(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (4-x)*p(n-1, x) - p(n-2, x), p(0, x) = 1, p(1, x) = 4-x.

From G. C. Greubel, Aug 20 2023: (Start)

T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).

Sum_{k=0..n} T(n, k) = A001906(n+1).

Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n).

T(n, n) = (-1)^n.

T(n, n-1) = 4*A181983(n), n >= 1.

T(n, n-2) = (-1)^n*A139278(n-1), n >= 2.

T(n, 0) = A001353(n+1). (End)