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 A240313 #8 Dec 25 2023 17:38:26
%S A240313 1,1,1,1,3,3,5,5,8,11,15,19,27,32,43,53,70,84,112,135,174,212,268,324,
%T A240313 407,490,606,731,897,1075,1312,1567,1899,2265,2726,3238,3886,4598,
%U A240313 5486,6482,7698,9063,10727,12592,14846,17391,20427,23862,27952,32568,38033
%N A240313 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (maximal part of p).
%F A240313 a(n) = A240312(n) + A240314(n) for n >= 0.
%F A240313 a(n) + A240310(n) = A000041(n) for n >= 0.
%e A240313 a(6) counts these 5 partitions: 3111, 222, 2211, 21111, 111111.
%t A240313 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
%t A240313 Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
%t A240313 Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
%t A240313 Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
%t A240313 Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
%t A240313 Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)
%Y A240313 Cf. A240310, A240311, A240312, A240314, A000041, A118053.
%K A240313 nonn,easy
%O A240313 0,5
%A A240313 _Clark Kimberling_, Apr 05 2014