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 A240311 #8 Dec 25 2023 17:38:04
%S A240311 1,1,1,2,4,5,8,10,16,22,32,42,59,76,103,134,179,228,301,382,494,623,
%T A240311 796,995,1259,1564,1957,2419,3005,3690,4552,5562,6815,8288,10095,
%U A240311 12218,14808,17842,21514,25823,30999,37058,44306,52775,62851,74613,88538,104764
%N A240311 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (maximal part of p).
%F A240311 a(n) = A240310(n) + A240312(n) for n >= 0.
%F A240311 a(n) + A240314(n) = A000041(n) for n >= 0.
%e A240311 a(6) counts these 8 partitions: 6, 51, 42, 411, 33, 3111, 321, 2211.
%t A240311 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
%t A240311 Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
%t A240311 Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
%t A240311 Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
%t A240311 Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
%t A240311 Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)
%Y A240311 Cf. A240310, A240312, A240313, A240314, A000041, A118052.
%K A240311 nonn,easy
%O A240311 0,4
%A A240311 _Clark Kimberling_, Apr 05 2014