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A136600
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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}},.
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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; internal format)
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OFFSET
| 1,2
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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]
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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, Graduaste Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978
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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
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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}
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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]
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CROSSREFS
| Cf. A129844.
Sequence in context: A115237 A105542 A200057 * A136672 A097750 A133544
Adjacent sequences: A136597 A136598 A136599 * A136601 A136602 A136603
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 24 2008
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