A387017 Expansion of (Product_{k>=1} (1 - x^k)^2/(1 - 5*x^k + x^(2*k)) - 1)/3.

1, 6, 28, 139, 660, 3192, 15260, 73254, 350848, 1681650, 8056608, 38604748, 184963130, 886226880, 4246152960, 20344613659, 97476826932, 467039887908, 2237722185188, 10721572793580, 51370139753240, 246129134364792, 1179275522335680, 5650248517615128

Offset: 1

Author

Christian Kassel, Aug 13 2025

Keywords

• nonn

Comments

a(n) is the value at q = (5 + sqrt(21))/2 of C_n(q)/(q^{n-1}(q - 1)^2), where C_n(q) is the number of codimension n ideals of the algebra of two-variable Laurent polynomials over a finite field of order q. The number C_n(q) is a palindromic polynomial of degree 2n with integer coefficients in the variable q and it is divisible by (q-1)^2.

Links

• Christian Kassel and Christophe Reutenauer, Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables, arXiv:1505.07229 [math.AG], 2015-2016; Michigan Math. J. 67 (2018), 715-741.
• Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016; The Ramanujan Journal 46 (2018), 633-655.
• Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025.

Crossrefs

Cf. A030221, A386706, A329156.

Programs

Mathematica

nmax = 25; Rest[CoefficientList[Series[(Product[(1 - x^k)^2/(1 - 5*x^k + x^(2*k)), {k, 1, nmax}] - 1)/3, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 14 2025 *)

Formula

G.f.: (Product_{k>=1} (1 - x^k)^2/(1 - 5*x^k + x^(2*k)) - 1)/3

a(2^k) = A030221(2^k-1). (Follows from Cor. 4.5 of Kassel and Reutenauer (2025).)

a(n) ~ (3 + sqrt(21))^(2*n-1) / (2^(2*n-1) * 3^n). - Vaclav Kotesovec, Aug 14 2025