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 1-4 of 4 results.

A363068 Number of partitions p of n such that (1/5)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 59, 73, 94, 117, 148, 181, 228, 277, 344, 418, 514, 621, 762, 917, 1116, 1342, 1624, 1945, 2348, 2803, 3366, 4012, 4798, 5700, 6798, 8052, 9565, 11305, 13383, 15771, 18618, 21880, 25745, 30187, 35414, 41414, 48461, 56531, 65967
Offset: 0

Views

Author

Seiichi Manyama, May 16 2023

Keywords

Comments

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

Examples

			a(8) = 2 counts these partitions:  521, 5111.

Links

Crossrefs

Cf. A002865, A238479, A363066, A363067.
Cf. A237827, A363047.
Cf. A035297, A035298.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(6*k)/Product[1 - x^j, {j, 1, 5*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^(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^(6*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)
  • PARI
    a(n) = sum(k=0, n\6, #partitions(n-6*k, 5*k));

Formula

G.f.: Sum_{k>=0} x^(6*k)/Product_{j=1..5*k} (1-x^j).
a(n) ~ Gamma(1/5) * Pi^(1/5) * exp(Pi*sqrt(2*n/3)) / (25 * 2^(21/10) * 3^(3/5) * n^(11/10)). - Vaclav Kotesovec, Jun 19 2025

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

A035296 Expansion of sum ( q^n / product( 1-q^k, k=1..4*n), n=0..inf ).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 18, 28, 41, 60, 85, 119, 164, 225, 304, 408, 542, 716, 938, 1222, 1582, 2037, 2609, 3326, 4220, 5332, 6708, 8407, 10497, 13061, 16197, 20020, 24671, 30313, 37141, 45383, 55311, 67242, 81552, 98678, 119135, 143522, 172545, 207018, 247899, 296294, 353492
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A363067, A385068.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 4*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^(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^k/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)

Formula

a(n) ~ Gamma(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(29/8) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jun 17 2025

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025

A238591 Number of partitions p of n such that 4*min(p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 45, 60, 81, 109, 144, 190, 247, 320, 412, 529, 675, 854, 1078, 1355, 1695, 2117, 2626, 3251, 4010, 4932, 6047, 7394, 9012, 10959, 13290, 16083, 19407, 23379, 28090, 33689, 40317, 48158, 57406, 68324, 81155, 96248
Offset: 1

Views

Author

Clark Kimberling, Mar 01 2014

Keywords

Examples

			a(9) = 5 counts these partitions:  441, 4311, 4221, 42111, 411111.

Links

Crossrefs

Cf. A117989, A238589, A238590.
Cf. A237826, A363067.

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-5*i, i), i=1..n/5):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 03 2014
  • Mathematica
    Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Min[p]]], {n, 50}]
(* Second program: *)
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 - 5*i, i], {i, 1, n/5}];
Array[a, 60] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Formula

G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023
From Vaclav Kotesovec, Jun 19 2025: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 97*Pi/(24*sqrt(6))) / sqrt(n)).
A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(3/2)). (End)
Showing 1-4 of 4 results.

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