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 A241409 #4 Apr 28 2014 18:42:07
%S A241409 0,0,1,1,2,3,3,5,6,9,12,16,23,26,39,45,67,78,106,130,171,207,270,329,
%T A241409 419,516,637,787,978,1190,1451,1775,2166,2613,3173,3827,4613,5537,
%U A241409 6659,7948,9523,11316,13505,16014,19059,22455,26667,31376,37079,43501,51282
%N A241409 Number of partitions of n such that the number of parts having multiplicity >1 is a part and the number of distinct parts is a part.
%C A241409 As used here, the term "distinct parts" includes each number, once, that occurs more than once; e.g., the distinct parts of the partition {4,3,3,1,1,1} are 4, 3, 1.
%e A241409 a(6) counts these 3 partitions: 411, 3111, 21111.
%t A241409 z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
%t A241409 Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}] (* A241408 *)
%t A241409 Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241409 *)
%t A241409 Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
%t A241409 Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411 *)
%t A241409 Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412 *)
%Y A241409 Cf. A241408, A241410, A241411, A241412.
%K A241409 nonn,easy
%O A241409 0,5
%A A241409 _Clark Kimberling_, Apr 22 2014