A386706 Expansion of ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.

0, 1, 5, 18, 71, 260, 990, 3672, 13775, 51343, 191860, 715770, 2672298, 9972092, 37220040, 138903480, 518408351, 1934712530, 7220497115, 26947209762, 100568547820, 375326739216, 1400739172470, 5227629044040, 19509779871450, 72811487038701, 271736178975820, 1014133216234068

Offset: 0

Author

Christian Kassel, Jul 30 2025

Keywords

• nonn

Comments

a(n) is the value at q = 2 + sqrt(3) 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

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• 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. A001834, A329156, A387017.

Programs

Mathematica

nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^2/(1 - 4*x^k + x^(2*k)), {k, 1, nmax}] - 1)/2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 30 2025 *)

Formula

G.f.: ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.

a(2^k) = A001834(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 of Kassel-Reutenauer paper "Pairs of intertwined integer sequences".

a(n) ~ (1 + sqrt(3))^(2*n-1) / 2^n. - Vaclav Kotesovec, Jul 30 2025

Extensions

a(0)=0 added, offset changed to 0, a(7) corrected and more terms added by Vaclav Kotesovec, Jul 30 2025