A124037 Triangle read by rows: row n gives coefficients of increasing powers of x in characteristic polynomial of the matrix (-1)^n*M_n, where M_n is the tridiagonal matrix defined in the Comments line.

1, 1, -1, 2, -4, 1, 5, -13, 7, -1, 13, -40, 33, -10, 1, 34, -120, 132, -62, 13, -1, 89, -354, 483, -308, 100, -16, 1, 233, -1031, 1671, -1345, 595, -147, 19, -1, 610, -2972, 5561, -5398, 3030, -1020, 203, -22, 1, 1597, -8495, 17984, -20410, 13893, -5943, 1610, -268, 25, -1, 4181, -24110, 56886, -73816, 59059

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

Keywords

• sign
• tabl

Comments

The matrices M_n for n=1, 2, 3, ... are:

1 X 1 {{1}},

2 X 2 {{1, -1}, {-1, 3}},

3 X 3 {{1, -1, 0}, {-1, 3, -1}, {0, -1, 3}},

4 X 4 {{1, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 3}},

5 X 5 {{1, -1, 0, 0, 0}, {-1, 3, -1, 0, 0}, {0, -1, 3, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 3}},

6 X 6 {{1, -1, 0, 0, 0, 0}, {-1, 3, -1, 0, 0, 0}, {0, -1, 3, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, { 0, 0, 0, -1, 3, -1}, {0, 0, 0, 0, -1, 3}}, ...

Subtriangle of the triangle given by (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2014

Riordan array ((1-2*x)/(1-3*x+x^2), -x/(1-3*x+x^2)). - Philippe Deléham, Mar 04 2014

Examples

			Triangle begins:
{1},
{1, -1},
{2, -4, 1},
{5, -13, 7, -1},
{13, -40, 33, -10, 1},
{34, -120,132, -62, 13, -1},
{89, -354, 483, -308, 100, -16, 1},
For example, the characteristic polynomial of M_3 is x^3-7*x^2+13*x-5, so row 3 is 5, -13, 7, -1.
Triangle (0, 1, 1, 1, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, ...) begins:
1;
0, 1;
0, 1, -1;
0, 2, -4, 1;
0, 5, -13, 7, -1;
0, 13, -40, 33, -10, 1;
0, 34, -120, 132, -62, 13, -1;
0, 89, -354, 483, -308, 100, -16, 1; - _Philippe Deléham_, Mar 04 2014
		

Crossrefs

Cf. A126126.

Programs

Mathematica

T[n_, m_, d_] := If[ n == m && n > 1 && m > 1, 3, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[ d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]

Sage

@CachedFunction
def T(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = T(n-1,k) if n==1 else 3*T(n-1,k)
    return T(n-1,k-1) - T(n-2,k) - h
A124037 = lambda n,k: (-1)^n*T(n,k)
for n in (0..9): [A124037(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

T(n,k) = 3*T(n-1,k) - T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 1, T(1,1) = -1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 04 2014

G.f.: -(-1+2*x)/(1-3*x+x^2+x*y). - R. J. Mathar, Aug 11 2015

Extensions

Edited by N. J. A. Sloane, Mar 02 2008