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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.

%I A136678 #13 Jan 27 2014 22:53:02
%S A136678 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 A136678 -112,54,-12,1,-4,-38,190,-293,210,-77,14,-1,-7,-40,314,-664,659,-352,
%U A136678 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 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.
%C A136678 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 A136678 For n=4, M is the Cartan matrix of type F_4.
%C A136678 Row sums are 1, 1, -1, -2, -2, 0, 1, 1, -1, -2, -2, ...
%F A136678 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 A136678 Triangle begins:
%e A136678 {1},
%e A136678 {2, -1},
%e A136678 {2, -4, 1},
%e A136678 {2, -9, 6, -1},
%e A136678 {1, -16, 20, -8,1},
%e A136678 {0, -24, 50, -35, 10, -1},
%e A136678 {-2, -32, 104, -112, 54, -12, 1},
%e A136678 {-4, -38, 190, -293, 210, -77, 14, -1},
%e A136678 {-7, -40, 314, -664, 659, -352, 104, -16, 1},
%e A136678 {-10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1},
%e A136678 {-14, -20,677, -2512, 4234, -3992, 2274, -800, 170, -20, 1}
%e A136678 ...
%e A136678 For n=4, the matrix M is
%e A136678 [2,-1,0,0],
%e A136678 [-1,2,-1,0],
%e A136678 [0,-2,2,-1],
%e A136678 [0,0,-1,2],
%e A136678 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 A136678 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 A136678 tabl,sign
%O A136678 0,2
%A A136678 _Roger L. Bagula_, Apr 05 2008
%E A136678 Edited by _N. J. A. Sloane_, Jan 27 2014