A136678 - OEIS

%I #13 Jan 27 2014 22:53:02

%S 1,2,-1,2,-4,1,2,-9,6,-1,1,-16,20,-8,1,0,-24,50,-35,10,-1,-2,-32,104,

%T -112,54,-12,1,-4,-38,190,-293,210,-77,14,-1,-7,-40,314,-664,659,-352,

%U 104,-16,1,-10,-35,478,-1349,1772,-1286,546,-135,18,-1,-14,-20,677,-2512,4234,-3992,2274,-800,170,-20,1

%N 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.

%C 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.

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

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

%F 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]]]

%e Triangle begins:

%e {1},

%e {2, -1},

%e {2, -4, 1},

%e {2, -9, 6, -1},

%e {1, -16, 20, -8,1},

%e {0, -24, 50, -35, 10, -1},

%e {-2, -32, 104, -112, 54, -12, 1},

%e {-4, -38, 190, -293, 210, -77, 14, -1},

%e {-7, -40, 314, -664, 659, -352, 104, -16, 1},

%e {-10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1},

%e {-14, -20,677, -2512, 4234, -3992, 2274, -800, 170, -20, 1}

%e ...

%e For n=4, the matrix M is

%e [2,-1,0,0],

%e [-1,2,-1,0],

%e [0,-2,2,-1],

%e [0,0,-1,2],

%e 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.

%t 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];

%K tabl,sign

%O 0,2

%A _Roger L. Bagula_, Apr 05 2008

%E Edited by _N. J. A. Sloane_, Jan 27 2014