Christian Kassel - cp's OEIS frontend

User: Christian Kassel

Christian Kassel has authored 2 sequences.

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

Christian Kassel, Aug 13 2025

nonn

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.

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.

Cf. A030221, A386706, A329156.

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 *)

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

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

Original entry on oeis.org

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

Christian Kassel, Jul 30 2025

nonn

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.

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.

Cf. A001834, A329156, A387017.

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 *)

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

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

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User: Christian Kassel

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

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

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cp's OEIS Frontend

User: Christian Kassel

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

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

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