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Search: id:A124037
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| A124037 |
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Triangle read by rows: row n gives coefficients of increasing powers of x in characteristic polynomial of the matrix (-1)^n*M_n, where M_n is the tridiagonal matrix defined in the Comments line. |
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+0
1
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| 1, 1, -1, 2, -4, 1, 5, -13, 7, -1, 13, -40, 33, -10, 1, 34, -120, 132, -62, 13, -1, 89, -354, 483, -308, 100, -16, 1, 233, -1031, 1671, -1345, 595, -147, 19, -1, 610, -2972, 5561, -5398, 3030, -1020, 203, -22, 1, 1597, -8495, 17984, -20410, 13893, -5943, 1610, -268, 25, -1, 4181, -24110, 56886, -73816, 59059
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OFFSET
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1,4
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COMMENT
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The matrices M_n for n=1, 2, 3, ... are: 1 X 1 {{1}}, 2 X 2 {{1, -1}, {-1, 3}}, 3 X 3 {{1, -1, 0}, {-1, 3, -1}, {0, -1, 3}}, 4 X 4 {{1, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 3}},
5 X 5 {{1, -1, 0, 0, 0}, {-1, 3, -1, 0, 0}, {0, -1, 3, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 3}}, 6 X 6 {{1, -1, 0, 0, 0, 0}, {-1, 3, -1, 0, 0, 0}, {0, -1, 3, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, { 0, 0, 0, -1, 3, -1}, {0, 0, 0, 0, -1, 3}}, ...
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EXAMPLE
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Triangle begins:
{1},
{1, -1},
{2, -4, 1},
{5, -13, 7, -1},
{13, -40, 33, -10, 1},
{34, -120,132, -62, 13, -1},
{89, -354, 483, -308, 100, -16, 1},
For example, the characteristic polynomial of M_3 is x^3-7*x^2+13*x-5, so row 3 is 5, -13, 7, -1.
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MATHEMATICA
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T[n_, m_, d_] := If[ n == m && n > 1 && m > 1, 3, If[n == m - 1 || n == m + 1, -1, If[n == m == 1, 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[ d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]
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CROSSREFS
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Sequence in context: A080427 A118906 A085059 this_sequence A090285 A047908 A125847
Adjacent sequences: A124034 A124035 A124036 this_sequence A124038 A124039 A124040
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KEYWORD
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sign
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Nov 03 2006
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 02 2008
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