A304176 Number of partitions of n^3 into exactly n parts.

1, 1, 4, 61, 1906, 91606, 6023602, 505853354, 51900711796, 6306147384659, 886745696653253, 141778041323736643, 25417656781153090889, 5052180112449982704619, 1103058286595668300801794, 262487324530101028337614478, 67628783852463631751658038290

Offset

0,3

Links

Chai Wah Wu, Table of n, a(n) for n = 0..136 (terms 0..96 from Alois P. Heinz)

Formula

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

Example

n | Partitions of n^3 into exactly n parts
--+------------------------------------------------------------
1 |   1.
2 |   7+1 = 6+2 = 5+3 = 4+4.
3 |   25+ 1+1 = 24+ 2+1 = 23+ 3+1 = 23+ 2+2 = 22+ 4+1 = 22+ 3+2
  | = 21+ 5+1 = 21+ 4+2 = 21+ 3+3 = 20+ 6+1 = 20+ 5+2 = 20+ 4+3
  | = 19+ 7+1 = 19+ 6+2 = 19+ 5+3 = 19+ 4+4 = 18+ 8+1 = 18+ 7+2
  | = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2 = 17+ 7+3 = 17+ 6+4
  | = 17+ 5+5 = 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 = 15+ 6+6
  | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6
  | = 13+13+1 = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6
  | = 13+ 7+7 = 12+12+3 = 12+11+4 = 12+10+5 = 12+ 9+6 = 12+ 8+7
  | = 11+11+5 = 11+10+6 = 11+ 9+7 = 11+ 8+8 = 10+10+7 = 10+ 9+8
  | =  9+ 9+9.

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):
seq(a(n), n=0..20);  # Alois P. Heinz, May 07 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];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Prog

(PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^3-n)))), n^3-n)}
(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 A304176(n): return b(n**3-n, n) # Chai Wah Wu, Sep 09 2021, after Alois P. Heinz

Crossrefs

Cf. A206240, A238608, A304208, A304212.

Sequence in context: A252973 A262654 A218472 * A326225 A102193 A232156

Adjacent sequences: A304173 A304174 A304175 * A304177 A304178 A304179

Keyword

nonn

Author

Seiichi Manyama, May 07 2018

Status

approved