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.

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).

A347606 Number of partitions of n^n into exactly n parts.

Original entry on oeis.org

0, 1, 2, 61, 117874, 33219689231, 2559960025059106420, 85975912953927216830024650654, 1841153609473379088124269084031755459049386
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2021

Keywords

Crossrefs

Cf. A206240, A304176, A347605, A347607.

Programs

  • PARI
    a(n) = polcoef(prod(k=1, n, 1/(1-x^k+x*O(x^(n^n-n)))), n^n-n);

Formula

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

A347617 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 into exactly n parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 4, 7, 1, 0, 1, 1, 8, 61, 34, 1, 0, 1, 1, 16, 547, 1906, 192, 1, 0, 1, 1, 32, 4921, 117874, 91606, 1206, 1, 0, 1, 1, 64, 44287, 7478386, 53830967, 6023602, 8033, 1, 0, 1, 1, 128, 398581, 477568114, 33219689231, 43054503928, 505853354, 55974, 1, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2021

Keywords

Examples

			Square array begins:
  0, 1,   1,     1,        1,           1, ...
  1, 1,   1,     1,        1,           1, ...
  0, 1,   2,     4,        8,          16, ...
  0, 1,   7,    61,      547,        4921, ...
  0, 1,  34,  1906,   117874,     7478386, ...
  0, 1, 192, 91606, 53830967, 33219689231, ...

Crossrefs

Columns k=0..3 give A063524, A000012, A206240, A304176.
Main diagonal gives A347606.
Cf. A238016, A347615, A347618.

Programs

  • PARI
    T(n, k) = if(k==0, n==1, polcoef(prod(j=1, n, 1/(1-x^j+x*O(x^(n^k-n)))), n^k-n));

Formula

T(n,k) = [x^(n^k-n)] Product_{j=1..n} 1/(1-x^j).

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 1-5 of 5 results.

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