A196019 Hodge structure on relative homology of some varieties related to cluster algebras of type A.

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

2,5

Comments

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

Links

Table of n, a(n) for n=2..56.

Formula

G.f.: G(q,x) = x + Sum_{n>1} g_n(q)*x^n satisfies x=G-G^2/(1-q*G^2)/(1-q*G)/(1+G).

Example

Triangle begins:
  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;
  ...
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.

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 // Rest // Flatten (* Jean-François Alcover, Mar 17 2020 *)

Crossrefs

Cf. A171711.

Sequence in context: A389263 A136267 A109960 * A056940 A168288 A157523

Adjacent sequences: A196016 A196017 A196018 * A196020 A196021 A196022

Keyword

nonn,tabl

Author

F. Chapoton, Sep 26 2011

Extensions

1st row of length 1 dropped to conform to the "tabl" triangle shape by Andrey Zabolotskiy, Mar 27 2025

Status

approved