A136678 Triangle read by rows: n-th row (n>=0) gives coefficients of characteristic polynomial of n X n generalized Cartan matrix M defined in Comments.

1, 2, -1, 2, -4, 1, 2, -9, 6, -1, 1, -16, 20, -8, 1, 0, -24, 50, -35, 10, -1, -2, -32, 104, -112, 54, -12, 1, -4, -38, 190, -293, 210, -77, 14, -1, -7, -40, 314, -664, 659, -352, 104, -16, 1, -10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1, -14, -20, 677, -2512, 4234, -3992, 2274, -800, 170, -20, 1

Offset: 0

Author

Roger L. Bagula, Apr 05 2008

Keywords

• tabl
• sign

Comments

M has 2's on the main diagonal, -1's on the two adjacent diagonals, except that the entry M[floor(n/2)+1, floor(n/2)] is -2.

For n=4, M is the Cartan matrix of type F_4.

Row sums are 1, 1, -1, -2, -2, 0, 1, 1, -1, -2, -2, ...

Examples

			Triangle begins:
{1},
{2, -1},
{2, -4, 1},
{2, -9, 6, -1},
{1, -16, 20, -8,1},
{0, -24, 50, -35, 10, -1},
{-2, -32, 104, -112, 54, -12, 1},
{-4, -38, 190, -293, 210, -77, 14, -1},
{-7, -40, 314, -664, 659, -352, 104, -16, 1},
{-10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1},
{-14, -20,677, -2512, 4234, -3992, 2274, -800, 170, -20, 1}
...
For n=4, the matrix M is
[2,-1,0,0],
[-1,2,-1,0],
[0,-2,2,-1],
[0,0,-1,2],
which has charpoly x^4-8*x^3+20*x^2-16*x+1. The coefficients (starting with the constant term) give row 4 of the triangle.
		

Programs

Mathematica

T[n_, m_, d_] := If[ n == m, 2, If[n == Floor[d/2] + 1 && m == Floor[d/2], -2, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a0 = Table[M[d], {d, 1, 10}]; Table[Det[M[d]], {d, 1, 10}]; g = Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] MatrixForm[a];

Formula

T(n, m, d)= If[ n == m, 2, If[n == Floor[d/2] + 1 && m == Floor[d/2], -2, If[(n == m - 1 || n == m + 1), -1, 0]]]

Extensions

Edited by N. J. A. Sloane, Jan 27 2014