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.

Previous Showing 11-13 of 13 results.

A304208 Number of partitions of n^3 into exactly n distinct parts.

Original entry on oeis.org

1, 1, 3, 48, 1425, 66055, 4234086, 348907094, 35277846729, 4236771148454, 590133028697501, 93613602614249377, 16671698429605679621, 3295006292978246618505, 715884159450254458674982, 169624990695197593491828744, 43538384149387312404895504349
Offset: 0

Views

Author

Seiichi Manyama, May 08 2018

Keywords

Examples

			n | Partitions of n^3 into exactly n distinct parts
--+-------------------------------------------------------------
1 |   1.
2 |   7+1 = 6+2 = 5+3.
3 |   24+ 2+1 = 23+ 3+1 = 22+ 4+1 = 22+ 3+2 = 21+ 5+1 = 21+ 4+2
  | = 20+ 6+1 = 20+ 5+2 = 20+ 4+3 = 19+ 7+1 = 19+ 6+2 = 19+ 5+3
  | = 18+ 8+1 = 18+ 7+2 = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2
  | = 17+ 7+3 = 17+ 6+4 = 16+10+1 = 16+ 9+2 = 16+ 8+3 = 16+ 7+4
  | = 16+ 6+5 = 15+11+1 = 15+10+2 = 15+ 9+3 = 15+ 8+4 = 15+ 7+5
  | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6
  | = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6 = 12+11+4
  | = 12+10+5 = 12+ 9+6 = 12+ 8+7 = 11+10+6 = 11+ 9+7 = 10+ 9+8.

Links

Crossrefs

Cf. A107379, A128854, A304176.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+b(n-i, min(i, n-i)))
    end:
a:= n-> b(n^3-n*(n+1)/2, n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 08 2018
  • Mathematica
    $RecursionLimit = 2000;
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1]+b[n-i, Min[i, n-i]]];
a[n_] :=  b[n^3 - n(n+1)/2, n];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^3-n*(n+1)/2)))), n^3-n*(n+1)/2)}

Formula

a(n) = [x^(n^3-n*(n+1)/2)] Product_{k=1..n} 1/(1-x^k).

A321239 a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k^2*m).

Original entry on oeis.org

1, 1, 3, 16, 141, 1534, 19111, 262103, 3853373, 59763670, 966945204, 16191250596, 278933800080, 4921604827876, 88627915588351, 1624349874930925, 30231112607904743, 570284342486800214, 10887435073866747752, 210086404047975194316, 4092940691144348506396, 80432925119259253535963
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2018

Keywords

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ..., a_n) to the equation 1^2*a_1 + 2^2*a_2 + ... + n^2*a_n = n^3.
Also the number of partitions of n^3 into square parts not greater than n^2. - Paul D. Hanna, Feb 02 2024

Examples

			1^2* 0 + 2^2*0 + 3^2*3 = 27.
1^2* 1 + 2^2*2 + 3^2*2 = 27.
1^2* 2 + 2^2*4 + 3^2*1 = 27.
1^2* 3 + 2^2*6 + 3^2*0 = 27.
1^2* 5 + 2^2*1 + 3^2*2 = 27.
1^2* 6 + 2^2*3 + 3^2*1 = 27.
1^2* 7 + 2^2*5 + 3^2*0 = 27.
1^2* 9 + 2^2*0 + 3^2*2 = 27.
1^2*10 + 2^2*2 + 3^2*1 = 27.
1^2*11 + 2^2*4 + 3^2*0 = 27.
1^2*14 + 2^2*1 + 3^2*1 = 27.
1^2*15 + 2^2*3 + 3^2*0 = 27.
1^2*18 + 2^2*0 + 3^2*1 = 27.
1^2*19 + 2^2*2 + 3^2*0 = 27.
1^2*23 + 2^2*1 + 3^2*0 = 27.
1^2*27 + 2^2*0 + 3^2*0 = 27.
So a(3) = 16.

Crossrefs

Cf. A001156, A107379, A321238.

Programs

  • PARI
    {a(n) = polcoeff(prod(i=1, n, sum(j=0, n^3\i^2, x^(i^2*j)+x*O(x^(n^3)))), n^3)}
  • PARI
    {a(n) = polcoeff( 1/prod(k=1,n, 1 - x^(k^2) +x*O(x^(n^3)) ), n^3) }
for(n=0,20, print1(a(n),", ")) \\ Paul D. Hanna, Feb 02 2024

A364055 Number of integer partitions of n satisfying (length) = (mean). Partitions of n into sqrt(n) parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0, 192, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1206, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8033, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55974, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Jul 07 2023

Keywords

Examples

			The a(0) = 1 through a(9) = 7 partitions:
  ()  (1)  .  .  (22)  .  .  .  .  (333)
                 (31)              (432)
                                   (441)
                                   (522)
                                   (531)
                                   (621)
                                   (711)

Crossrefs

The strict case is A107379(sqrt(n)).
Without zeros we have A206240.
These partitions have ranks A363951.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean, ranks A316413.
Cf. A025065, A026905, A237984, A327472, A327482, A363944, A363949.

Programs

  • Mathematica
    Table[Length[If[n==0,{{}},Select[IntegerPartitions[n],Mean[#]==Length[#]&]]],{n,0,30}]

Formula

a(n^2) = A206240(n).
Previous Showing 11-13 of 13 results.

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

Content is available under The OEIS End-User License Agreement.