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Search: id:A136674
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| A136674 |
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Triangular sequence made from matrices of the type( Cartan G_n types): M(3)= {{2, -1, 0}, {-1, 2, -1}, {0, -3, 2}} with polynomial recursion: p(x, n) = (2 - x)*p(x, n - 1) - p(x, n - 2). |
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+0
7
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| 1, 2, -1, 1, -4, 1, 0, -8, 6, -1, -1, -12, 19, -8, 1, -2, -15, 44, -34, 10, -1, -3, -16, 84, -104, 53, -12, 1, -4, -14, 140, -258, 200, -76, 14, -1, -5, -8, 210, -552, 605, -340, 103, -16, 1, -6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1, -7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums:
{1, 1, -2, -3, -1, 2, 3, 1, -2, -3, -1}
The 3 X 3 matrix with determinant zero is forbidden in the Cartan definition in the same way as E_9 is forbidden.
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FORMULA
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T(n, m, d) = If[ n == m, 2, If[n == d && m == d - 1, -3, If[(n == m - 1 || n == m + 1), -1, 0]]] or p(x, n) = (2 - x)*p(x, n - 1) - p(x, n - 2)
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EXAMPLE
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{1},
{2, -1},
{1, -4, 1},
{0, -8, 6, -1},
{-1, -12, 19, -8, 1},
{-2, -15, 44, -34, 10, -1},
{-3, -16, 84, -104, 53, -12, 1},
{-4, -14, 140, -258, 200, -76, 14, -1},
{-5, -8, 210, -552,605, -340, 103, -16, 1},
{-6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1},
{-7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1}
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MATHEMATICA
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(* tridiagonal matrix code*) T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -3, 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]; (* polynomial recursion: three initial terms necessary*) Clear[p] p[x, 0] = 1; p[x, 1] = (2 - x); p[x, 2] = 1 - 4 x + x^2; p[x_, n_] := p[x, n] = (2 - x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
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CROSSREFS
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Adjacent sequences: A136671 A136672 A136673 this_sequence A136675 A136676 A136677
Sequence in context: A010247 A087605 A106246 this_sequence A064645 A008307 A099238
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 05 2008
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