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%I #6 Dec 12 2015 04:33:09
%S 1,2,-1,4,-4,1,6,-11,6,-1,5,-20,21,-8,1,4,-34,56,-36,10,-1,3,-52,125,
%T -120,55,-12,1,2,-73,246,-329,220,-78,14,-1,1,-96,440,-784,714,-364,
%U 105,-16,1,0,-120,730,-1679,1992,-1364,560,-136,18,-1,-1,-144,1140,-3304,4949,-4356,2379,-816,171,-20,1,-2,-167,1694
%N Triangle of coefficients of characteristic polynomials of a special type of Cartan matrix: E_n for E_6,E_7,E_8,E_11 example M(6)/ E_6: {{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, -1}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 2}},.
%C Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0}.
%C Solution for a polynomial recursion gives for higher polynomials:
%C p1 = Join[{1}, Table[CharacteristicPolynomial[MO[n], x], {n, 1, 12}]];
%C Table[Solve[{p1[[n]] - (a0*x - b0)*p1[[n - 1]] - c0*p1[[n - 2]] == 0, p1[[n + 1]] - (a0*x - b0)* p1[[n]] - c0*p1[[n - 1]] == 0, p1[[n + 2]] - (a0*x - b0)*p1[[n + 1]] - c0*p1[[n]] == 0}, {a0, b0, c0}], {n, 3, 10}];
%C Polynomial recursion:
%C P[x, n] = (2 - x)*P[x, n - 1] + P[x, n - 2]
%D R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8.page 139
%D E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl, 1957
%D Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978
%F h(n,m)=If[ n == m, a[n], If[n == m - 1 ||n == m + 1 || n == m - 3 || n == m + 3, If[n == m - 1 && m < d,b[m - 1], If[n == m + 1 && n < d, b[n - 1], If[n ==m - 3 || n == m + 3, If[n == m - 3 && m == d, c[m - 3], If[n == m + 3 && n == d, c[n - 3], 0]]]]]]] ; for n,m<=d
%e {1},
%e {2, -1},
%e {4, -4, 1},
%e {6, -11, 6, -1},
%e {5, -20, 21, -8, 1},
%e {4, -34, 56, -36, 10, -1},
%e {3, -52, 125, -120,55, -12, 1},
%e {2, -73, 246, -329, 220, -78, 14, -1},
%e {1, -96, 440, -784, 714, -364, 105, -16, 1},
%e {0, -120, 730, -1679, 1992, -1364, 560, -136, 18, -1},
%e {-1, -144, 1140, -3304, 4949, -4356, 2379,-816, 171, -20, 1},
%e {-2, -167, 1694, -6069, 11210, -12297, 8554, -3875, 1140, -210, 22, -1},
%e {-3, -188, 2415, -10528, 23540, -31448, 27026, -15488, 5984, -1540, 253, -24, 1}
%t a[n_] := 2; b[n_] := -1; c[n_] := -1; T[n_, m_, d_] := If[ n == m, a[n], If[n == m - 1 || n == m + 1 || n ==m - 3 || n == m + 3, If[n == m - 1 &&m < d, b[m - 1], If[n == m + 1 && n < d, b[n - 1], If[n == m - 3 || n == m + 3, If[n == m - 3 && m == d, c[m - 3], If[n == m + 3 && n == d, c[n - 3], 0]]]]]]] MO[d_] := Table[If[TrueQ[T[n, m, d] == Null], 0, T[n, m, d]], {n, 1, d}, {m, 1, d}]; a1 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[MO[n], x], x], {n, 1, 12}]]' Flatten[a1]
%Y Cf. A129844.
%K uned,tabl,sign
%O 1,2
%A _Roger L. Bagula_, Mar 24 2008
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