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.

Showing 1-5 of 5 results.

A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 16, 20, 27, 33, 45, 55, 72, 89, 116, 142, 181, 222, 281, 343, 429, 522, 649, 786, 967, 1168, 1429, 1719, 2088, 2504, 3026, 3615, 4345, 5174, 6192, 7349, 8755, 10360, 12297, 14507, 17154, 20182, 23788, 27910, 32790, 38374, 44955, 52480, 61307, 71402
Offset: 0

Views

Author

Seiichi Manyama, May 16 2023

Keywords

Examples

			a(7) = 3 counts these partitions:  331, 3211, 31111.

Links

Crossrefs

Cf. A002865, A238479, A363067, A363068.
Cf. A008484, A237825, A363045.
Cf. A035295.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(4*k)/Product[1 - x^j, {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 18 2025 *)
nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(3*k))*(1-x^(3*k-1))*(1-x^(3*k-2))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(4*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)
Join[{1},Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]/3]&)],{n,60}]] (* _Harvey P. Dale, Jun 29 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, #partitions(n-4*k, 3*k));

Formula

G.f.: Sum_{k>=0} x^(4*k)/Product_{j=1..3*k} (1-x^j).
a(n) ~ Gamma(1/3) * Pi^(1/3) * exp(Pi*sqrt(2*n/3)) / (2^(13/6) * 3^(8/3) * n^(7/6)). - Vaclav Kotesovec, Jun 19 2025

A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 39, 51, 64, 81, 102, 128, 159, 198, 245, 304, 374, 460, 563, 689, 841, 1023, 1242, 1505, 1819, 2195, 2642, 3173, 3804, 4551, 5435, 6477, 7707, 9151, 10850, 12843, 15175, 17902, 21089, 24802, 29132, 34164, 40012, 46796, 54663, 63766
Offset: 0

Views

Author

Seiichi Manyama, May 16 2023

Keywords

Examples

			a(8) = 3 counts these partitions:  431, 4211, 41111.

Links

Crossrefs

Cf. A002865, A238479, A363066, A363068.
Cf. A237826, A363046.
Cf. A035296.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(5*k)/Product[1 - x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 18 2025 *)
nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(4*k))*(1-x^(4*k-1))*(1-x^(4*k-2))*(1-x^(4*k-3))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(5*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)
  • PARI
    a(n) = sum(k=0, n\5, #partitions(n-5*k, 4*k));

Formula

G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..4*k} (1-x^j).
a(n) ~ Gamma(1/4) * Pi^(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(49/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Jun 19 2025

A035297 Expansion of sum ( q^n / product( 1-q^k, k=1..5*n), n=0..inf ).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 19, 29, 43, 63, 90, 127, 176, 241, 327, 439, 585, 773, 1015, 1322, 1714, 2208, 2831, 3610, 4585, 5794, 7297, 9149, 11433, 14233, 17665, 21846, 26943, 33123, 40614, 49656, 60565, 73671, 89414, 108254, 130785, 157649, 189654, 227671, 272802, 326236, 389446
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A363068, A385069.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 16 2025 *)
nmax = 50; p=1; s=1; Do[p=Expand[p*(1-x^(5*k))*(1-x^(5*k-1))*(1-x^(5*k-2))*(1-x^(5*k-3))*(1-x^(5*k-4))];p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]];s+=x^k/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)

Formula

a(n) ~ Gamma(1/5) * exp(Pi*sqrt(2*n/3)) / (5 * 2^(8/5) * 3^(1/10) * Pi^(4/5) * n^(3/5)). - Vaclav Kotesovec, Jun 17 2025

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025

A035298 Expansion of sum ( q^n / product( 1-q^k, k=1..6*n), n=0..inf ).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 19, 30, 44, 65, 93, 132, 183, 253, 343, 462, 616, 816, 1071, 1399, 1813, 2339, 2999, 3828, 4861, 6149, 7743, 9714, 12140, 15120, 18766, 23220, 28640, 35224, 43199, 52838, 64458, 78441, 95226, 115336, 139381, 168077, 202258, 242900, 291140, 348300, 415922
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In general, for m>=1, if g.f. = Sum_{k>=0} x^k / Product_{j=1..m*k} (1 - x^j), then a(n) ~ Gamma(1/m) * exp(Pi*sqrt(2*n/3)) / (m * 2^((3*m + 1)/(2*m)) * 3^(1/(2*m)) * Pi^(1 - 1/m) * n^((m+1)/(2*m))). - Vaclav Kotesovec, Jun 17 2025

Links

Crossrefs

Cf. A363068, A385070.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 16 2025 *)
nmax = 50; p=1; s=1; Do[p=Expand[p*(1-x^(6*k))*(1-x^(6*k-1))*(1-x^(6*k-2))*(1-x^(6*k-3))*(1-x^(6*k-4))*(1-x^(6*k-5))];p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]];s+=x^k/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)

Formula

a(n) ~ Gamma(1/6) * exp(Pi*sqrt(2*n/3)) / (2^(31/12) * 3^(13/12) * Pi^(5/6) * n^(7/12)). - Vaclav Kotesovec, Jun 17 2025

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025

A361459 Number of partitions p of n such that 5*min(p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 23, 31, 44, 58, 82, 105, 142, 185, 244, 312, 409, 516, 664, 837, 1063, 1328, 1674, 2074, 2588, 3194, 3952, 4847, 5964, 7270, 8884, 10786, 13104, 15832, 19147, 23027, 27709, 33203, 39776, 47476, 56661, 67382, 80108, 94960, 112494, 132919, 156965
Offset: 1

Views

Author

Seiichi Manyama, May 17 2023

Keywords

Comments

From Vaclav Kotesovec, Jun 19 2025: (Start)
In general, for m>1, if g.f. = Sum_{k>=0} x^(m*k) / Product_{j>=k} (1 - x^j), then A000041(n) - a(n) ~ Pi * (m-1) * exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*n^(3/2)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + (m - 23/24)*Pi / sqrt(6)) / sqrt(n)). (End)

Links

Crossrefs

Cf. A117989, A238589, A238590, A238591.
Cf. A237827, A363068.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-6*i, i), i=1..n/6):
seq(a(n), n=1..60);  # Alois P. Heinz, May 17 2023
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n - 6 i, i], {i, 1, n/6}];
Array[a, 60] (* Jean-François Alcover, May 30 2024, after Alois P. Heinz *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k)/prod(j=k, N, 1-x^j))))

Formula

G.f.: Sum_{k>=1} x^(6*k)/Product_{j>=k} (1-x^j).
From Vaclav Kotesovec, Jun 19 2025: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 121*Pi/(24*sqrt(6))) / sqrt(n)).
A000041(n) - a(n) ~ 5 * Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). (End)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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