A304706 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm.
1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 6, 6, 8, 7, 11, 10, 11, 12, 15, 14, 18, 17, 20, 23, 27, 25, 31, 32, 35, 38, 43, 43, 51, 54, 59, 63, 71, 73, 85, 89, 96, 102, 113, 120, 134, 141, 149, 161, 175, 183, 203, 213, 233, 252, 280, 293, 319, 338, 360, 383, 409, 430, 468, 493, 531, 565
Offset: 0
Keywords
Examples
n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m) --+-----------------------------+--------------------------------------------- 1 | (1) | (1) 2 | (2) | (2) | (1, 1) | (1, 1/2) 3 | (3) | (3) | (1, 1, 1) | (1, 1/2, 1/3) 4 | (4) | (4) | (2, 2) | (2, 1) | (1, 1, 1, 1) | (1, 1/2, 1/3, 1/4) 5 | (5) | (5) | (2, 3) | (2, 3/2) | (1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5) 6 | (6) | (6) | (3, 3) | (3, 3/2) | (2, 2, 2) | (2, 1, 2/3) | (1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6) 7 | (7) | (7) | (3, 4) | (3, 2) | (1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7) 8 | (8) | (8) | (3, 5) | (3, 5/2) | (4, 4) | (4, 2/1) | (2, 3, 3) | (2, 3/2, 1) | (2, 2, 2, 2) | (2, 1, 2/3, 1/2) | (1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8) 9 | (9) | (9) | (4, 5) | (4, 5/2) | (2, 3, 4) | (2, 3/2, 4/3) | (3, 3, 3) | (3, 3/2, 1) | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)
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, 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, 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) <= A304705(n).