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%I #15 Nov 23 2020 08:02:58
%S 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,
%T 32,35,38,43,43,51,54,59,63,71,73,85,89,96,102,113,120,134,141,149,
%U 161,175,183,203,213,233,252,280,293,319,338,360,383,409,430,468,493,531,565
%N Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm.
%F a(n) <= A304705(n).
%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, 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, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5)
%e 6 | (6) | (6)
%e | (3, 3) | (3, 3/2)
%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 | (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 | (4, 5) | (4, 5/2)
%e | (2, 3, 4) | (2, 3/2, 4/3)
%e | (3, 3, 3) | (3, 3/2, 1)
%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, n+1, 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, 1];
%t a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%Y Cf. A053251, A053282, A304705, A304707, A304708.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 17 2018
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