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 A136600 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}},. 0
 1, 2, -1, 4, -4, 1, 6, -11, 6, -1, 5, -20, 21, -8, 1, 4, -34, 56, -36, 10, -1, 3, -52, 125, -120, 55, -12, 1, 2, -73, 246, -329, 220, -78, 14, -1, 1, -96, 440, -784, 714, -364, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0}. Solution for a polynomial recursion gives for higher polynomials: p1 = Join[{1}, Table[CharacteristicPolynomial[MO[n], x], {n, 1, 12}]]; 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}]; Polynomial recursion: P[x, n] = (2 - x)*P[x, n - 1] + P[x, n - 2] REFERENCES R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8.page 139 E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl, 1957 Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978 LINKS FORMULA 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 EXAMPLE {1}, {2, -1}, {4, -4, 1}, {6, -11, 6, -1}, {5, -20, 21, -8, 1}, {4, -34, 56, -36, 10, -1}, {3, -52, 125, -120,55, -12, 1}, {2, -73, 246, -329, 220, -78, 14, -1}, {1, -96, 440, -784, 714, -364, 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, -6069, 11210, -12297, 8554, -3875, 1140, -210, 22, -1}, {-3, -188, 2415, -10528, 23540, -31448, 27026, -15488, 5984, -1540, 253, -24, 1} MATHEMATICA 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] CROSSREFS Cf. A129844. Sequence in context: A105542 A208907 A200057 * A136672 A097750 A304623 Adjacent sequences:  A136597 A136598 A136599 * A136601 A136602 A136603 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula, Mar 24 2008 STATUS approved

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Last modified April 14 07:59 EDT 2021. Contains 342946 sequences. (Running on oeis4.)