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))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 07 2018
STATUS
approved