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Search: id:A112712
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| A112712 |
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Expansion of x/(1-x+2x^2-2x^3+2x^4-x^5+x^6). |
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+0
2
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| 0, 1, 1, -1, -1, 1, 0, -2, 0, 2, 0, -1, 1, 1, -1, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, 1, 0, -2, 0, 2, 0, -1, 1, 1, -1, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, 1, 0, -2, 0, 2, 0, -1, 1, 1, -1, -1, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,8
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COMMENT
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A modified Chebyshev transform of the Fibonacci numbers F(n) under the mapping g(x)->(1/(1+x^2)^2)g(x/(1+x^2)).
Characteristic polynomial obtained from the Jones polynomial for the link L6a2: f(x) = -1/x^(3/2) + 1/x^(5/2) - 2/x^(7/2) + 2/x^(9/2) - 2/x^(11/2) + 1/x^(13/2) - 1/x^(15/2). Characteristic polynomial when fractional powers are removed: p(x)=-1/(1 - x + 2 x^2 - 2 x^3 + 2 x^4 - x^5 + x^6). - Roger L. Bagula (rlbagulatfttn(AT)yahoo.com), Jun 06 2007
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REFERENCES
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L6a2: http://katlas.math.toronto.edu/wiki/Image:L6a2.gif
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FORMULA
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G.f.: x(1+x-x^2-x^3+x^4-2x^6+2x^8-x^10+x^11+x^12-x^13-x^14)/(1-x^20); a(n)=a(n-1)-2a(n-2)+2a(n-3)-2a(n-4)+a(n-5)-a(n-6); also a(n)=a(n-20); a(n)=sum{k=0..floor((n+2)/2), (-1)^(k+1)C(n-k+2, k-1)F(n-2k+2)}; a(n)=sum{k=0..n, F(k)*(-1)^((n-k)/2)*sum{j=0..n, C((j+k)/2, k)(1+(-1)^(n-j))(1+(-1)^(j-k))/4}}.
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MATHEMATICA
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f[x_] = -1/x^(3/2) + 1/x^(5/2) - 2/x^(7/2) + 2/x^(9/2) - 2/x^(11/2) + 1/x^(13/2) - 1/x^(15/2); p[x] = ExpandAll[FullSimplify[x^(3/2)/f[x]]/x^9]; Table[SeriesCoefficient[Series[p[x], {x, 0, 30}], n], {n, 0, 30}] - Roger L. Bagula (rlbagulatfttn(AT)yahoo.com), Jun 06 2007
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CROSSREFS
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Cf. A112713.
Adjacent sequences: A112709 A112710 A112711 this_sequence A112713 A112714 A112715
Sequence in context: A126211 A095414 A001877 this_sequence A026608 A026612 A046922
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KEYWORD
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easy,sign
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Sep 15 2005
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