OFFSET
0,4
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
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
EXAMPLE
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
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]
PROG
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Gary W. Adamson and Roger L. Bagula, Nov 03 2006
EXTENSIONS
Edited by N. J. A. Sloane, Mar 02 2008
STATUS
approved