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 A304705 #29 Nov 23 2020 08:02:32 %S A304705 1,1,2,3,3,4,6,5,6,8,9,9,12,11,14,17,16,17,23,22,27,31,30,33,40,41,46, %T A304705 50,54,57,70,70,77,88,92,99,111,115,129,142,152,160,175,183,199,223, %U A304705 234,255,283,299,328,347,370,390,430,455,489,523,557,592,642,674,724,784 %N A304705 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and 0 < d1 <= d2 <= ... <= dm. %H A304705 Alois P. Heinz, <a href="/A304705/b304705.txt">Table of n, a(n) for n = 0..400</a> %e A304705 n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m) %e A304705 --+-----------------------------+--------------------------------------------- %e A304705 1 | (1) | (1) %e A304705 2 | (2) | (2) %e A304705 | (1, 1) | (1, 1/2) %e A304705 3 | (3) | (3) %e A304705 | (1, 2) | (1, 1) %e A304705 | (1, 1, 1) | (1, 1/2, 1/3) %e A304705 4 | (4) | (4) %e A304705 | (2, 2) | (2, 1) %e A304705 | (1, 1, 1, 1) | (1, 1/2, 1/3, 1/4) %e A304705 5 | (5) | (5) %e A304705 | (2, 3) | (2, 3/2) %e A304705 | (1, 2, 2) | (1, 1, 2/3) %e A304705 | (1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5) %e A304705 6 | (6) | (6) %e A304705 | (2, 4) | (2, 2) %e A304705 | (3, 3) | (3, 3/2) %e A304705 | (1, 2, 3) | (1, 1, 1) %e A304705 | (2, 2, 2) | (2, 1, 2/3) %e A304705 | (1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6) %e A304705 7 | (7) | (7) %e A304705 | (3, 4) | (3, 2) %e A304705 | (2, 2, 3) | (2, 1, 1) %e A304705 | (1, 2, 2, 2) | (1, 1, 2/3, 1/2) %e A304705 | (1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7) %e A304705 8 | (8) | (8) %e A304705 | (3, 5) | (3, 5/2) %e A304705 | (4, 4) | (4, 2/1) %e A304705 | (2, 3, 3) | (2, 3/2, 1) %e A304705 | (2, 2, 2, 2) | (2, 1, 2/3, 1/2) %e A304705 | (1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8) %e A304705 9 | (9) | (9) %e A304705 | (3, 6) | (3, 3) %e A304705 | (4, 5) | (4, 5/2) %e A304705 | (2, 3, 4) | (2, 3/2, 4/3) %e A304705 | (3, 3, 3) | (3, 3/2, 1) %e A304705 | (1, 2, 3, 3) | (1, 1, 1, 3/4) %e A304705 | (1, 2, 2, 2, 2) | (1, 1, 2/3, 1/2, 2/5) %e A304705 | (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 A304705 b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0, %p A304705 b(n, r, i+1, t)+`if`(i/t>r, 0, b(n-i, i/t, i, t+1)))) %p A304705 end: %p A304705 a:= n-> b(n$2, 1$2): %p A304705 seq(a(n), n=0..80); # _Alois P. Heinz_, May 17 2018 %t A304705 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 A304705 a[n_] := b[n, n, 1, 1]; %t A304705 a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *) %Y A304705 Cf. A053251, A053282, A304706, A304707, A304708. %K A304705 nonn %O A304705 0,3 %A A304705 _Seiichi Manyama_, May 17 2018