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%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
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