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Search: id:A140882
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| A140882 |
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A set of Cartan like Martices with the properties that either the rows or columns as sums are zero that give a triangle of coefficients of characteristic polynomials: Example matrix 3x3(row sums zero): {{2, -2, 0}, {-1, 2, -1}, {0, -2, 2}}. |
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| 1, 2, -1, 0, -4, 1, 0, -8, 6, -1, 0, -12, 19, -8, 1, 0, -16, 44, -34, 10, -1, 0, -20, 85, -104, 53, -12, 1, 0, -24, 146, -259, 200, -76, 14, -1, 0, -28, 231, -560, 606, -340, 103, -16, 1, 0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1, 0, -36, 489, -1968, 3630, -3652, 2171, -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 are:
{1, 1, -3, -3, 0, 3, 3, 0, -3, -3, 0}.
This sequence of matrices was inspired by the Kemeny "dominant","hybrid"
and "recessive" matrix of genetic characteristics:
{{2, 2, 0},
{1, 2, 1},
{0, 2, 2}}/2^2
That type matrix has row sums equal one.
I noticed that it resembled an unsigned Cartan matrix of a D_n or B_n type
with rows sums zero.
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REFERENCES
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Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Printice - Hall, N, J., Section 3, Chapter VII, page 407
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FORMULA
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m(d)=If[ n == m, 2, If[(n == d &&m == d - 1) || (n == 1 && m == 2), -2, If[(n == m - 1 || n == m + 1), -1, 0]]; out_n,m=Coefficients(CharacteristicPolynomial(m(n))
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EXAMPLE
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{1},
{2, -1},
{0, -4, 1},
{0, -8, 6, -1},
{0, -12, 19, -8,1},
{0, -16, 44, -34, 10, -1},
{0, -20, 85, -104, 53, -12, 1},
{0, -24, 146, -259, 200, -76, 14, -1},
{0, -28, 231, -560, 606, -340, 103, -16, 1},
{0, -32, 344, -1092,1572, -1210, 532, -134, 18, -1},
{0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1}
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MATHEMATICA
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T[n_, m_, d_] := If[ n == m, 2, If[(n == d && m == d - 1) || ( n == 1 && m == 2), -2, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]
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CROSSREFS
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Sequence in context: A019094 A134082 A139360 this_sequence A143724 A143425 A166555
Adjacent sequences: A140879 A140880 A140881 this_sequence A140883 A140884 A140885
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jul 22 2008
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