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 A241086 #9 May 11 2025 18:53:19
%S A241086 0,1,1,1,1,2,2,2,2,3,4,4,5,5,6,7,8,9,11,12,14,15,17,19,21,24,27,31,34,
%T A241086 38,42,47,51,57,62,70,77,85,93,104,114,125,137,150,164,180,196,214,
%U A241086 234,255,279,304,332,360,393,426,464,502,545,589,640,691,749
%N A241086 Number of partitions p of n into distinct parts such that max(p) <= 2*(number of parts of p).
%F A241086 a(15) counts these 7 partitions: 8421, 7521, 7431, 654, 6531, 6432, 54321.
%t A241086 z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A241086 Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}] (* A241085 *)
%t A241086 Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
%t A241086 Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
%t A241086 Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
%t A241086 Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}] (* A241089 *)
%Y A241086 Cf. A241085, A241087, A241088, A241089.
%K A241086 nonn,easy
%O A241086 0,6
%A A241086 _Clark Kimberling_, Apr 17 2014