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 A240312 #6 Dec 25 2023 17:38:13
%S A240312 1,1,0,0,2,1,2,0,2,3,5,5,9,7,11,11,18,15,28,27,41,43,62,64,91,96,127,
%T A240312 140,184,200,260,287,365,410,511,573,717,803,985,1120,1359,1538,1859,
%U A240312 2106,2522,2870,3407,3872,4586,5207,6128,6976,8167,9284,10844,12321
%N A240312 Number of partitions p of n such that (maximal multiplicity of the parts of p) = (maximal part of p).
%F A240312 a(n) = A240311(n) - A240310(n) for n >= 0.
%F A240312 a(n) + A240310(n) + A240314(n) = A000041(n) for n >= 0.
%e A240312 a(6) counts these 2 partitions: 3111, 2211.
%t A240312 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
%t A240312 Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
%t A240312 Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
%t A240312 Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
%t A240312 Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
%t A240312 Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)
%Y A240312 Cf. A240310, A240311, A240313, A240314, A000041.
%K A240312 nonn,easy
%O A240312 0,5
%A A240312 _Clark Kimberling_, Apr 05 2014