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A143606
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Coefficient expansion sequence of symmetric polynomial: p(x)=1 + x - x^3 - x^5 - x^6 - x^7 - x^9 + x^11 + x^12.
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0
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1, -1, 1, 0, -1, 3, -3, 3, 0, -2, 6, -7, 8, -3, -1, 11, -16, 22, -15, 6, 18, -35, 56, -49, 33, 20, -68, 130, -138, 121, -13, -108, 279, -356, 374, -177, -102, 544, -847, 1037, -743, 162, 905, -1850, 2646, -2414, 1367, 1035, -3637, 6265, -6876
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| This polynomial is the lowest result save a factor of Lehmer's polynomial in a 12th degree symmetrical polynomial census. (Not Salem but nearly.)
Vector matrix Markov that gives the same sequence is:
CompanionMatrix[p_, x_] := Module[{cl = CoefficientList[p, x], deg,
m}, cl = Drop[cl/Last[cl], -1]; deg = Length[cl]; If[deg == 1, {-cl},
m = RotateLeft[IdentityMatrix[deg]]; m[[ -1]] = -cl; Transpose[m]]];
M = Transpose[CompanionMatrix[1 + x - x^3 - x^5 - x^6 - x^7 - x^9 + x^11 +
x^12, x]];
v[0] = Table[a[[n]], {n, 1, 12}];
v[n_] := v[n] = M.v[n - 1];
Table[v[n][[1]], {n, 0, 50}]
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FORMULA
| p(x)=1 + x - x^3 - x^5 - x^6 - x^7 - x^9 + x^11 + x^12; a(n)=Coefficient_expansion(x^12*p(1/x))
G.f.: x/(1+x-x^3-x^5-x^6-x^7-x^9+x^11+x^12). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 25 2009]
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MATHEMATICA
| f[x_] = 1 + x - x^3 - x^5 - x^6 - x^7 - x^9 + x^11 + x^12; g[x] = ExpandAll[x^12*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}];
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CROSSREFS
| Sequence in context: A196544 A002073 A130719 * A126660 A202698 A164884
Adjacent sequences: A143603 A143604 A143605 * A143607 A143608 A143609
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KEYWORD
| sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 26 2008
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