cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A335754 a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 1 modulo 3.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 17, 23, 30, 39, 51, 66, 84, 107, 135, 168, 209, 259, 319, 391, 478, 581, 703, 849, 1022, 1226, 1466, 1748, 2078, 2465, 2917, 3443, 4055, 4765, 5588, 6540, 7640, 8908, 10368, 12047, 13973, 16182, 18712, 21604, 24906, 28673, 32964, 37846, 43397
Offset: 0

Views

Author

Jeremy Lovejoy, Jun 20 2020

Keywords

Examples

			The 9 overpartitions counted by a(5) are: [5'], [4,1], [4,1'], [4',1], [4',1'], [2',1,1,1], [2',1',1,1], [1,1,1,1,1], [1',1,1,1,1].

Links

  • J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.

Crossrefs

Cf. A003105, A015128, A098151, A335755.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k-1)) * (1 + x^(3*k-2)) / (1 - x^(3*k-2)), {k, 1, nmax/3}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

Formula

G.f.: Product_{n>=1} (1+q^(3*n-1))*(1+q^(3*n-2))/(1-q^(3*n-2)).
a(n) ~ Gamma(1/3) * exp(2*Pi*sqrt(n)/3) / (2^(3/2) * sqrt(3) * Pi^(2/3) * n^(2/3)). - Vaclav Kotesovec, Jan 14 2021

A335755 a(n) is the number of overpartitions of n where overlined parts are not divisible by 3 and non-overlined parts are congruent to 2 modulo 3.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 9, 11, 14, 19, 24, 31, 39, 48, 61, 75, 93, 114, 139, 169, 205, 248, 298, 358, 428, 510, 607, 719, 851, 1005, 1182, 1389, 1628, 1904, 2225, 2592, 3015, 3501, 4058, 4698, 5429, 6264, 7216, 8302, 9538, 10944, 12541, 14351, 16403
Offset: 0

Views

Author

Jeremy Lovejoy, Jun 20 2020

Keywords

Examples

			The 6 overpartitions counted by a(6) are: [5,1'], [5',1'], [4',2], [4',2'], [2,2,2], [2',2,2].

Links

  • J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.

Crossrefs

Cf. A003105, A015128, A098151, A335754.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k-1)) * (1 + x^(3*k-2)) / (1 - x^(3*k-1)), {k, 1, nmax/3}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

Formula

G.f.: Product_{n>=1} (1+q^(3*n-1))*(1+q^(3*n-2))/(1-q^(3*n-1)).
a(n) ~ Pi^(2/3) * exp(2*Pi*sqrt(n)/3) / (3*sqrt(2)*Gamma(1/3)*n^(5/6)). - Vaclav Kotesovec, Jan 14 2021

A380481 Number of partitions of n into distinct parts less than n and not a multiple of 3.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 9, 11, 14, 15, 17, 20, 22, 25, 30, 33, 37, 42, 46, 52, 60, 66, 73, 82, 90, 101, 114, 125, 138, 153, 168, 186, 207, 227, 249, 274, 300, 330, 364, 398, 435, 476, 519, 568, 622, 678, 738, 804, 874, 952, 1038, 1127, 1223, 1327, 1438, 1561, 1694, 1834, 1984, 2146, 2320, 2509, 2714, 2930, 3161
Offset: 0

Views

Author

Harman Kaur, Jun 23 2025

Keywords

Examples

			a(3) = 1: [2,1].
a(5) = 1: [4,1].
a(6) = 2: [5,1], [4,2].
a(7) = 2: [5,2], [4,2,1].
a(8) = 2: [7,1], [5,2,1].
a(9) = 3: [8,1], [7,2], [5,4].
a(10) = 3: [8,2], [7,2,1], [5,4,1].
a(11) = 4: [10,1], [8,2,1], [7,4], [5,4,2].

Crossrefs

Cf. A003105, A011655, A357457.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n, n-1):
seq(a(n), n=0..74);  # Alois P. Heinz, Jul 24 2025
  • Mathematica
    CoefficientList[
 Series[-q/QPochhammer[q, q, 1] + q^3/QPochhammer[q^3, q, 1] +
    QPochhammer[-q, q^3]*QPochhammer[-q^2, q^3], {q, 0, 500}], q]

Formula

G.f.: x^3/(1-x^3) - x/(1-x) + Product_{n>=0} (1+x^(3n+1))*(1+x^(3n+2)).
a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Jul 25 2025
a(n) = A003105(n) - A011655(n). - Alois P. Heinz, Jul 25 2025
Previous Showing 41-43 of 43 results.

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