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 A240314 #6 Dec 25 2023 17:38:37
%S A240314 0,0,1,1,1,2,3,5,6,8,10,14,18,25,32,42,52,69,84,108,133,169,206,260,
%T A240314 316,394,479,591,713,875,1052,1280,1534,1855,2215,2665,3169,3795,4501,
%U A240314 5362,6339,7525,8868,10486,12324,14521,17020,19990,23366,27361,31905
%N A240314 Number of partitions p of n such that (maximal multiplicity of the parts of p) > (maximal part of p).
%F A240314 a(n) = A240313(n) - A240312(n) for n >= 0.
%F A240314 a(n) + A240310(n) + A240312(n) = A000041(n) for n >= 0.
%e A240314 a(6) counts these 3 partitions: 222, 2111, 111111.
%t A240314 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
%t A240314 Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
%t A240314 Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
%t A240314 Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
%t A240314 Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
%t A240314 Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)
%Y A240314 Cf. A240310, A240311, A240312, A240313, A000041.
%K A240314 nonn,easy
%O A240314 0,6
%A A240314 _Clark Kimberling_, Apr 05 2014