This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A304708 #14 Nov 23 2020 08:10:54 %S A304708 1,1,1,1,1,2,1,2,2,3,2,3,3,3,5,5,4,5,6,6,7,8,8,9,10,12,11,13,13,16,16, %T A304708 15,18,21,22,26,25,28,31,33,33,35,39,41,46,47,50,53,59,63,68,74,77,84, %U A304708 90,93,98,105,111,119,129,132,138,149,157,169,178,189,201,211,227 %N A304708 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and d1 < d2 < ... < dm. %F A304708 a(n) <= A304707(n). %e A304708 n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m) %e A304708 --+-----------------------------+------------------------- %e A304708 1 | (1) | (1) %e A304708 2 | (2) | (2) %e A304708 3 | (3) | (3) %e A304708 4 | (4) | (4) %e A304708 5 | (5) | (5) %e A304708 | (2, 3) | (2, 3/2) %e A304708 6 | (6) | (6) %e A304708 7 | (7) | (7) %e A304708 | (3, 4) | (3, 2) %e A304708 8 | (8) | (8) %e A304708 | (3, 5) | (3, 5/2) %e A304708 9 | (9) | (9) %e A304708 | (4, 5) | (4, 5/2) %e A304708 | (2, 3, 4) | (2, 3/2, 4/3) %p A304708 b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0, %p A304708 b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i+1, t+1)))) %p A304708 end: %p A304708 a:= n-> b(n, n+1, 1$2): %p A304708 seq(a(n), n=0..80); # _Alois P. Heinz_, May 17 2018 %t A304708 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 A304708 a[n_] := b[n, n + 1, 1, 1]; %t A304708 a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *) %Y A304708 Cf. A053251, A053282, A304705, A304706, A304707. %K A304708 nonn %O A304708 0,6 %A A304708 _Seiichi Manyama_, May 17 2018