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 A239737 #19 Apr 24 2014 10:27:39
%S A239737 0,1,1,2,2,5,7,10,13,21,28,38,54,77,99,137,180,236,306,398,504,644,
%T A239737 807,1018,1278,1599,1972,2458,3039,3743,4592,5659,6884,8436,10235,
%U A239737 12445,15021,18204,21842,26334,31501,37746,44956,53707,63657,75738,89536,106057
%N A239737 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part or the number of numbers having multiplicity > 1 is a part.
%F A239737 a(n) + A241417(n) = A000041(n) for n >= 0.
%e A239737 a(6) counts these 7 partitions: 42, 411, 321, 3111, 2211, 21111, 111111.
%t A239737 z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
%t A239737 Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}] (* A241413 *)
%t A239737 Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}] (* A241414 *)
%t A239737 Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
%t A239737 Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
%t A239737 Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
%t A239737 Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)
%Y A239737 Cf. A241413, A241414, A241415, A241416, A241417, A000041.
%K A239737 nonn,easy
%O A239737 0,4
%A A239737 _Clark Kimberling_, Apr 23 2014