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 A304706 #15 Nov 23 2020 08:02:58 %S A304706 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 A304706 32,35,38,43,43,51,54,59,63,71,73,85,89,96,102,113,120,134,141,149, %U A304706 161,175,183,203,213,233,252,280,293,319,338,360,383,409,430,468,493,531,565 %N A304706 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm. %F A304706 a(n) <= A304705(n). %e A304706 n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m) %e A304706 --+-----------------------------+--------------------------------------------- %e A304706 1 | (1) | (1) %e A304706 2 | (2) | (2) %e A304706 | (1, 1) | (1, 1/2) %e A304706 3 | (3) | (3) %e A304706 | (1, 1, 1) | (1, 1/2, 1/3) %e A304706 4 | (4) | (4) %e A304706 | (2, 2) | (2, 1) %e A304706 | (1, 1, 1, 1) | (1, 1/2, 1/3, 1/4) %e A304706 5 | (5) | (5) %e A304706 | (2, 3) | (2, 3/2) %e A304706 | (1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5) %e A304706 6 | (6) | (6) %e A304706 | (3, 3) | (3, 3/2) %e A304706 | (2, 2, 2) | (2, 1, 2/3) %e A304706 | (1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6) %e A304706 7 | (7) | (7) %e A304706 | (3, 4) | (3, 2) %e A304706 | (1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7) %e A304706 8 | (8) | (8) %e A304706 | (3, 5) | (3, 5/2) %e A304706 | (4, 4) | (4, 2/1) %e A304706 | (2, 3, 3) | (2, 3/2, 1) %e A304706 | (2, 2, 2, 2) | (2, 1, 2/3, 1/2) %e A304706 | (1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8) %e A304706 9 | (9) | (9) %e A304706 | (4, 5) | (4, 5/2) %e A304706 | (2, 3, 4) | (2, 3/2, 4/3) %e A304706 | (3, 3, 3) | (3, 3/2, 1) %e A304706 | (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 A304706 b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0, %p A304706 b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i, t+1)))) %p A304706 end: %p A304706 a:= n-> b(n, n+1, 1$2): %p A304706 seq(a(n), n=0..80); # _Alois P. Heinz_, May 17 2018 %t A304706 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 A304706 a[n_] := b[n, n + 1, 1, 1]; %t A304706 a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *) %Y A304706 Cf. A053251, A053282, A304705, A304707, A304708. %K A304706 nonn %O A304706 0,3 %A A304706 _Seiichi Manyama_, May 17 2018