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A137396
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Triangular sequence from the coefficients of a recursive polynomials set that is matched to the chromatic polynomial of cyclic graphs: p(x, n) = (-2 + x)*p(x, n - 1) + (-1 + x)*p(x, n - 2).
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0
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1, -1, 1, 0, -1, 1, 0, 2, -3, 1, 0, -3, 6, -4, 1, 0, 4, -10, 10, -5, 1, 0, -5, 15, -20, 15, -6, 1, 0, 6, -21, 35, -35, 21, -7, 1, 0, -7, 28, -56, 70, -56, 28, -8, 1, 0, 8, -36, 84, -126, 126, -84, 36, -9, 1, 0, -9, 45, -120, 210, -252, 210, -120, 45, -10, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Row sums are: {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
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REFERENCES
| Weisstein, Eric W. "Chromatic Polynomial." http://mathworld.wolfram.com/ChromaticPolynomial.html
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FORMULA
| p(x, n) = (-2 + x)*p(x, n - 1) + (-1 + x)*p(x, n - 2).
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EXAMPLE
| {1},
{-1, 1},
{0, -1, 1},
{0, 2, -3,1},
{0, -3, 6, -4, 1},
{0, 4, -10, 10, -5, 1},
{0, -5, 15, -20, 15, -6, 1},
{0, 6, -21, 35, -35, 21, -7, 1},
{0, -7, 28, -56, 70, -56, 28, -8, 1},
{0, 8, -36, 84, -126, 126, -84, 36, -9, 1},
{0, -9, 45, -120, 210, -252, 210, -120, 45, -10, 1}
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MATHEMATICA
| (* recursive polynomials*) Clear[p, x, a] p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = -1 + x; p[x, 2] = -x + x^2; p[x, 3] = 2 x - 3 x^2 + x^3; p[x_, n_] := p[x, n] = (-2 + x)*p[x, n - 1] + (-1 + x)*p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] (* MathWorld software *) Clear[a]; << MathWorld`Graphs` Table[ExpandAll[ChromaticPolynomial[Cycle[n], z]], {n, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[ChromaticPolynomial[Cycle[n], z], z], {n, 1, 10}]]; Flatten[a]
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CROSSREFS
| Sequence in context: A004572 A082839 A130717 * A178245 A167666 A115352
Adjacent sequences: A137393 A137394 A137395 * A137397 A137398 A137399
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 10 2008
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