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

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

Offset: 0

Author

Seiichi Manyama, May 08 2018

Keywords

• nonn

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

• Alois P. Heinz, Table of n, a(n) for n = 0..100

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).