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A196019
Hodge structure on relative homology of some varieties related to cluster algebras of type A.
0
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
OFFSET
1,6
COMMENTS
This is a refinement of the Euler characteristics of the same spaces, given by seq. A171711
FORMULA
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).
EXAMPLE
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
MAPLE
eq:=x-G+G**2/(1-q*G**2)/(1-q*G)/(1+G); solu:=solve(eq, G); taylor(solu, x, 8)
MATHEMATICA
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 *)
CROSSREFS
Cf. A171711.
Sequence in context: A008957 A136267 A109960 * A056940 A168288 A157523
KEYWORD
nonn,tabl
AUTHOR
F. Chapoton, Sep 26 2011
STATUS
approved