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 A240310 #6 Dec 25 2023 17:37:53
%S A240310 0,0,1,2,2,4,6,10,14,19,27,37,50,69,92,123,161,213,273,355,453,580,
%T A240310 734,931,1168,1468,1830,2279,2821,3490,4292,5275,6450,7878,9584,11645,
%U A240310 14091,17039,20529,24703,29640,35520,42447,50669,60329,71743,85131,100892
%N A240310 Number of partitions p of n such that (maximal multiplicity of the parts of p) < (maximal part of p).
%F A240310 a(n) = A240311(n) - A240312(n) for n >= 0.
%F A240310 a(n) + A240312(n) + A240314(n) = A000041(n) for n >= 0.
%e A240310 a(6) counts these 6 partitions: 6, 51, 42, 411, 33, 321.
%t A240310 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
%t A240310 Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
%t A240310 Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
%t A240310 Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
%t A240310 Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
%t A240310 Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314*)
%Y A240310 Cf. A240311, A240312, A240313, A240314, A000041.
%K A240310 nonn,easy
%O A240310 0,4
%A A240310 _Clark Kimberling_, Apr 05 2014