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 1-6 of 6 results.

A347607 Number of partitions of n^n.

Original entry on oeis.org

1, 1, 5, 3010, 365749566870782, 8630901377559029573671524821295260243701883575513498104067
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2021

Keywords

Comments

The next term a(6) = 1.30449952...*10^235 is too large to include.
a(7) = 1.5782589391...*10^1004. - Chai Wah Wu, Sep 09 2021

Links

Crossrefs

Main diagonal of A347615.
Cf. A000041, A000312, A064682, A072213, A128854.

Programs

  • Maple
    a:= n-> combinat[numbpart](n^n):
seq(a(n), n=0..6);  # Alois P. Heinz, Sep 09 2021
  • PARI
    a(n) = numbpart(n^n);
  • Python
    from sympy.functions import partition
def A347607(n): return partition(n**n) # Chai Wah Wu, Sep 09 2021

Formula

a(n) = A000041(n^n).

A347615 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 22, 30, 5, 1, 1, 1, 231, 3010, 231, 7, 1, 1, 1, 8349, 18004327, 1741630, 1958, 11, 1, 1, 1, 1741630, 133978259344888, 365749566870782, 3163127352, 17977, 15, 1, 1, 1, 4351078600, 233202632378520643600875145, 61847822068260244309086870983975, 1606903190858354689128371, 15285151248481, 173525, 22, 1
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2021

Keywords

Examples

			Square array begins:
  1, 1,   1,       1,               1, ...
  1, 1,   1,       1,               1, ...
  1, 2,   5,      22,             231, ...
  1, 3,  30,    3010,        18004327, ...
  1, 5, 231, 1741630, 365749566870782, ...

Crossrefs

Columns k=0..3 give A000012, A000041, A072213, A128854.
Rows n=0+1, 2-10 give A000012, A068413, A248728, A068413(2*n), A248730, A248732, A248734, A068413(3*n), A248728(2*n), A070177.
Main diagonal gives A347607.
Cf. A238016, A347617, A347618, A347621.

Programs

  • PARI
    T(n, k) = numbpart(n^k);

Formula

T(n,k) = A000041(n^k).

A281501 Number of partitions of n^3 into distinct parts.

Original entry on oeis.org

1, 1, 6, 192, 16444, 3207086, 1258238720, 916112394270, 1168225267521350, 2496696209705056142, 8635565795744155161506, 46977052491046305327286932, 392416122247953159916295467008, 4931628582570689013431218105121792, 91603865924570978521516549662581412000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2017

Keywords

Examples

			a(2) = 6 because we have [8], [7, 1], [6, 2], [5, 3], [5, 2, 1] and [4, 3, 1].

Links

Crossrefs

Cf. A000009, A000578, A072213, A072243, A128854.

Programs

  • Mathematica
    Table[PartitionsQ[n^3], {n, 0, 10}]

Formula

a(n) = [x^(n^3)] Product_{k>=1} (1 + x^k).
a(n) = A000009(A000578(n)).
a(n) ~ exp(Pi*n^(3/2)/sqrt(3))/(4*3^(1/4)*n^(9/4)).

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).

A304212 Number of partitions of n^3 into exactly n^2 parts.

Original entry on oeis.org

1, 1, 5, 318, 112540, 139620591, 491579082022, 4303961368154069, 85434752794871493882, 3588523098005804563697043, 302194941264401427042462944147, 48844693123353655726678707534158535, 14615188708581196626576773497618986350642
Offset: 0

Views

Author

Seiichi Manyama, May 08 2018

Keywords

Examples

			n | Partitions of n^3 into exactly n^2 parts
--+-------------------------------------------------
1 | 1.
2 | 5+1+1+1 = 4+2+1+1 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2.

Links

Crossrefs

Cf. 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^2, n^2):
seq(a(n), n=0..15);  # 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^2, n^2]; a /@ Range[0, 15] (* Jean-François Alcover, Nov 15 2020, after Alois P. Heinz *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n^2, 1/(1-x^k+x*O(x^(n^3-n^2)))), n^3-n^2)}
  • Python
    import sys
from functools import lru_cache
sys.setrecursionlimit(10**6)
@lru_cache(maxsize=None)
def b(n,i): return 1 if n == 0 or i == 1 else b(n,i-1)+b(n-i,min(i,n-i))
def A304212(n): return b(n**3-n**2,n**2) # Chai Wah Wu, Sep 09 2021, after Alois P. Heinz

Formula

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

A347604 Number of partitions of n^3 into n or more parts.

Original entry on oeis.org

1, 1, 21, 2996, 1741256, 3163112106, 15285150382556, 175943559746571618, 4453575699565108152534, 233202632378520005314974035, 24061467864032622392081524591073, 4700541557913558825449308701662220085, 1681375219875327721201831964319709743701981
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2021

Keywords

Crossrefs

Cf. A128854, A238608, A304176, A347585, A347605.

Formula

a(n) = [x^(n^3)] Sum_{k>=n} x^k / Product_{j=1..k} (1 - x^j).
a(n) = A128854(n) + A304176(n) - A238608(n).
Showing 1-6 of 6 results.

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