A304708 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and d1 < d2 < ... < dm.

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 3, 5, 5, 4, 5, 6, 6, 7, 8, 8, 9, 10, 12, 11, 13, 13, 16, 16, 15, 18, 21, 22, 26, 25, 28, 31, 33, 33, 35, 39, 41, 46, 47, 50, 53, 59, 63, 68, 74, 77, 84, 90, 93, 98, 105, 111, 119, 129, 132, 138, 149, 157, 169, 178, 189, 201, 211, 227

Offset: 0

Author

Seiichi Manyama, May 17 2018

Keywords

• nonn

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+-------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
3 | (3)                         | (3)
4 | (4)                         | (4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
6 | (6)                         | (6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
9 | (9)                         | (9)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
		

Crossrefs

Cf. A053251, A053282, A304705, A304706, A304707.

Programs

Maple

b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i+1, t+1))))
    end:
a:= n-> b(n, n+1, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

Mathematica

b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t >= r, 0, b[n - i, i/t, i + 1, t + 1]]]];
a[n_] := b[n, n + 1, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

a(n) <= A304707(n).