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A136678
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Triangular sequence based on matrices of an F_n Cartan-type ( exceptional group) with example: M(4)= {{2, -1, 0, 0}, {-1, 2, -1, 0}, {0, -2, 2, -1}, {0, 0, -1, 2}}.
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6
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1, 2, -1, 2, -4, 1, 2, -9, 6, -1, 1, -16, 20, -8, 1, 0, -24, 50, -35, 10, -1, -2, -32, 104, -112, 54, -12, 1, -4, -38, 190, -293, 210, -77, 14, -1, -7, -40, 314, -664, 659, -352, 104, -16, 1, -10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1, -14, -20, 677, -2512, 4234, -3992, 2274, -800, 170, -20, 1
(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, -2, -2, 0, 1, 1, -1, -2, -2}
No polynomial recursion was found, but the Gram-Schmidt analysis
shows that they are probably orthogonal to each other.
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FORMULA
| 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]]]
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EXAMPLE
| {1},
{2, -1},
{2, -4, 1},
{2, -9, 6, -1},
{1, -16, 20, -8,1},
{0, -24, 50, -35, 10, -1},
{-2, -32, 104, -112, 54, -12, 1},
{-4, -38, 190, -293, 210, -77, 14, -1},
{-7, -40, 314, -664, 659, -352, 104, -16, 1},
{-10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1},
{-14, -20,677, -2512, 4234, -3992, 2274, -800, 170, -20, 1}
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MATHEMATICA
| 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];
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CROSSREFS
| Sequence in context: A080246 A113413 A125694 * A110162 A199087 A087704
Adjacent sequences: A136675 A136676 A136677 * A136679 A136680 A136681
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 05 2008
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