A304705 - OEIS

%I #29 Nov 23 2020 08:02:32

%S 1,1,2,3,3,4,6,5,6,8,9,9,12,11,14,17,16,17,23,22,27,31,30,33,40,41,46,

%T 50,54,57,70,70,77,88,92,99,111,115,129,142,152,160,175,183,199,223,

%U 234,255,283,299,328,347,370,390,430,455,489,523,557,592,642,674,724,784

%N Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and 0 < d1 <= d2 <= ... <= dm.

%H Alois P. Heinz, <a href="/A304705/b304705.txt">Table of n, a(n) for n = 0..400</a>

%e n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)

%e --+-----------------------------+---------------------------------------------

%e 1 | (1)                         | (1)

%e 2 | (2)                         | (2)

%e   | (1, 1)                      | (1, 1/2)

%e 3 | (3)                         | (3)

%e   | (1, 2)                      | (1, 1)

%e   | (1, 1, 1)                   | (1, 1/2, 1/3)

%e 4 | (4)                         | (4)

%e   | (2, 2)                      | (2, 1)

%e   | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)

%e 5 | (5)                         | (5)

%e   | (2, 3)                      | (2, 3/2)

%e   | (1, 2, 2)                   | (1, 1, 2/3)

%e   | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)

%e 6 | (6)                         | (6)

%e   | (2, 4)                      | (2, 2)

%e   | (3, 3)                      | (3, 3/2)

%e   | (1, 2, 3)                   | (1, 1, 1)

%e   | (2, 2, 2)                   | (2, 1, 2/3)

%e   | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)

%e 7 | (7)                         | (7)

%e   | (3, 4)                      | (3, 2)

%e   | (2, 2, 3)                   | (2, 1, 1)

%e   | (1, 2, 2, 2)                | (1, 1, 2/3, 1/2)

%e   | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)

%e 8 | (8)                         | (8)

%e   | (3, 5)                      | (3, 5/2)

%e   | (4, 4)                      | (4, 2/1)

%e   | (2, 3, 3)                   | (2, 3/2, 1)

%e   | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)

%e   | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)

%e 9 | (9)                         | (9)

%e   | (3, 6)                      | (3, 3)

%e   | (4, 5)                      | (4, 5/2)

%e   | (2, 3, 4)                   | (2, 3/2, 4/3)

%e   | (3, 3, 3)                   | (3, 3/2, 1)

%e   | (1, 2, 3, 3)                | (1, 1, 1, 3/4)

%e   | (1, 2, 2, 2, 2)             | (1, 1, 2/3, 1/2, 2/5)

%e   | (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)

%p b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,

%p       b(n, r, i+1, t)+`if`(i/t>r, 0, b(n-i, i/t, i, t+1))))

%p     end:

%p a:= n-> b(n$2, 1$2):

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, May 17 2018

%t 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]]]];

%t a[n_] := b[n, n, 1, 1];

%t a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)

%Y Cf. A053251, A053282, A304706, A304707, A304708.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 17 2018