A304707 - OEIS

%I #15 Nov 23 2020 08:03:10

%S 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,

%T 20,23,22,26,30,29,32,35,34,37,43,44,48,54,54,59,67,70,76,81,84,89,97,

%U 101,110,119,123,129,139,145,155,171,176,189,201,211,228,245,257,274,295

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

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

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

%e 1 | (1)                         | (1)

%e 2 | (2)                         | (2)

%e 3 | (3)                         | (3)

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

%e 4 | (4)                         | (4)

%e 5 | (5)                         | (5)

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

%e 6 | (6)                         | (6)

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

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

%e 7 | (7)                         | (7)

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

%e 8 | (8)                         | (8)

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

%e 9 | (9)                         | (9)

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

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

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

%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+1, 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 + 1, 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, A304705, A304706, A304708.

%K nonn

%O 0,4

%A _Seiichi Manyama_, May 17 2018