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A196019
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Hodge structure on relative homology of some varieties related to cluster algebras of type A.
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0
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1, 1, 1, 1, 1, 5, 1, 1, 15, 9, 1, 1, 35, 50, 14, 1, 1, 70, 207, 113, 20, 1, 1, 126, 694, 672, 217, 27, 1, 1, 210, 1986, 3215, 1690, 376, 35, 1, 1, 330, 5028, 12969, 10484, 3663, 606, 44, 1, 1, 495, 11550, 45529, 54588, 28045, 7170, 925, 54, 1
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OFFSET
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1,6
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COMMENTS
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This is a refinement of the Euler characteristics of the same spaces, given by seq. A171711
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LINKS
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FORMULA
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G.f.: G(x) = Sum_{n>=1} g(n)*x^n satisfies x=G-G^2/(1-q*G^2)/(1-q*G)/(1+G).
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EXAMPLE
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The polynomial g(3)=1+q describes the weights of the relative homology of the punctured affine line A^1\{0} with respect to the divisor {1,2}. This is related to the cluster algebra of type A1.
1,
1,
1, 1,
1, 5, 1,
1, 15, 9, 1,
1, 35, 50, 14, 1,
1, 70, 207, 113, 20, 1,
1, 126, 694, 672, 217, 27, 1
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MAPLE
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eq:=x-G+G**2/(1-q*G**2)/(1-q*G)/(1+G); solu:=solve(eq, G); taylor(solu, x, 8)
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MATHEMATICA
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CoefficientList[#, q]& /@ ((G /. Solve[x - G + G^2/(1 - q G^2)/(1 - q G)/ (1 + G) == 0, G][[1]]) + O[x]^12 // CoefficientList[#, x]&) // Rest // Flatten (* Jean-François Alcover, Mar 17 2020 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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