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Search: id:A123969
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| A123969 |
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A triangular sequence from a Beraha type recursive polynomial using 5 X 5 n centered tridiagonal matrices with chromatic polynomial central roots to its characteristic polynomial. |
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+0
1
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| -1, -1, 1, 0, 4, -6, -4, 11, -6, 1, 6, -41, 75, -60, 74, -119, 57, 64, -93, 47, -11, 1, 144, -492, 886, -1076, 489, 618, -1063, 1154, -1672, 1618, -410, -682, 785, -392, 108, -16, 1, 744, -2567, 3782, -1075, -6736, 18095, -29241, 29006, -12952, -3601, 11554, -18942, 24467, -17741, 2907, 6473, -6678, 3357, -1026
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Recursive polynomial type from a classic paper related to Tutte-Beraha constants and the four color graph/ topology problem.
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REFERENCES
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Limits of zeros of recursively defined polynomials, S. Beraha, J. Kahane and N. J. Weiss, Proc Natl Acad Sci U S A. 1975 November; 72(11): 4209.
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FORMULA
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M(n)={{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, q(n,x)=CharacteristicPolynomial(M(n),x) p(k, x) = -Sum[q(n, x)*p(k - n, x), {n, 1, k - 1}]
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EXAMPLE
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Triangle begins:
{-1},
{-1, 1},
{0, 4, -6, -4, 11, -6, 1},
{6, -41,75, -60, 74, -119, 57, 64, -93, 47, -11, 1},
{144, -492, 886, -1076, 489, 618, -1063, 1154, -1672, 1618, -410, -682, 785, -392, 108, -16, 1}
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MATHEMATICA
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M[n_] := {{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, {0, 0, 0, -1, n}}; q[n_, x_] = Factor[CharacteristicPolynomial[M[n], x]]; p[0, x] = -1; p[1, x] = x - 1; p[k_, x_] := p[k, x] = -Sum[q[n, x]*p[k - n, x], {n, 1, k - 1}]; Table[Expand[p[n, x]], {n, 0, 10}] w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
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CROSSREFS
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Sequence in context: A114595 A143545 A143521 this_sequence A019188 A019244 A019189
Adjacent sequences: A123966 A123967 A123968 this_sequence A123970 A123971 A123972
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KEYWORD
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uned,probation,tabf,sign
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 29 2006
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