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Hodge structure on relative homology of some varieties related to cluster algebras of type A.
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%I #14 Mar 17 2020 10:18:21

%S 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,

%T 217,27,1,1,210,1986,3215,1690,376,35,1,1,330,5028,12969,10484,3663,

%U 606,44,1,1,495,11550,45529,54588,28045,7170,925,54,1

%N Hodge structure on relative homology of some varieties related to cluster algebras of type A.

%C This is a refinement of the Euler characteristics of the same spaces, given by seq. A171711

%F 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).

%e 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.

%e 1,

%e 1,

%e 1, 1,

%e 1, 5, 1,

%e 1, 15, 9, 1,

%e 1, 35, 50, 14, 1,

%e 1, 70, 207, 113, 20, 1,

%e 1, 126, 694, 672, 217, 27, 1

%p eq:=x-G+G**2/(1-q*G**2)/(1-q*G)/(1+G); solu:=solve(eq, G); taylor(solu, x, 8)

%t 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 *)

%Y Cf. A171711.

%K nonn,tabl

%O 1,6

%A _F. Chapoton_, Sep 26 2011