A329156 -id:A329156 - cp's OEIS frontend

A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

1, -1, -3, -3, -4, 3, 2, 19, 21, 32, 40, 45, 16, 8, -18, -125, -164, -291, -358, -530, -588, -724, -592, -675, -358, -207, 570, 1201, 2208, 3333, 4944, 6490, 8277, 10492, 11800, 13260, 14328, 14722, 12942, 12075, 5640, 603, -10444, -21120, -39360, -55876, -83488

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

sign

Convolution inverse of A329156.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003056, A032198, A077285, A088305, A104575, A319668, A329156, A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..46);  # Alois P. Heinz, Jul 18 2025

nmax = 46; CoefficientList[Series[Product[(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Product[(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k / (1 - x^k)^2).
G.f.: exp(-Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} 1 / B(x^k), where B(x) = g.f. of A088305.
a(n) = Sum_{k=0..A003056(n)} (-1)^k * A385001(n,k). - Alois P. Heinz, Jul 18 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

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

Original entry on oeis.org

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

A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j(2k - 1))).

Original entry on oeis.org

1, 1, 3, 9, 22, 59, 156, 405, 1061, 2786, 7284, 19071, 49948, 130738, 342288, 896175, 2346134, 6142287, 16080852, 42100020, 110219366, 288558380, 755455128, 1977807393, 5177967900, 13556094631, 35490316938, 92914858431, 243254253904, 636847905903, 1667289469704, 4365020491362

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Weigh transform of A032198.

Cf. A032198, A102186, A329156.

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(j (2 k - 1)), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^(2 k - 1)/(1 - x^(2 k - 1))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1 / (1 - x^(2*k - 1) / (1 - x^(2*k - 1))^2).
G.f.: Product_{k>=1} (1 + x^k)^A032198(k).
a(n) ~ c * phi^(2*n) / sqrt(5), where c = Product_{k>=2} 1/(1 - phi^(2 - 4*k)/(phi^(2 - 4*k) - 1)^2) = 1.07705428718361459418304978675229012... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019

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A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

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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 - 5*x^k + x^(2*k)) - 1)/3.

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A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j*(2*k - 1))).

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

A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j(2k - 1))).