%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