A304707 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and d1 < d2 < ... < dm.

1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 4, 5, 7, 5, 7, 8, 8, 10, 12, 10, 11, 14, 14, 14, 18, 17, 20, 23, 22, 26, 30, 29, 32, 35, 34, 37, 43, 44, 48, 54, 54, 59, 67, 70, 76, 81, 84, 89, 97, 101, 110, 119, 123, 129, 139, 145, 155, 171, 176, 189, 201, 211, 228, 245, 257, 274, 295

Offset

0,4

Links

Table of n, a(n) for n=0..69.

Example

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

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$2, 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];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Crossrefs

Cf. A053251, A053282, A304705, A304706, A304708.

Sequence in context: A302257 A324748 A320387 * A112218 A172366 A132148

Adjacent sequences: A304704 A304705 A304706 * A304708 A304709 A304710

Keyword

nonn

Author

Seiichi Manyama, May 17 2018

Status

approved