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 A241417 #4 Apr 29 2014 00:07:59
%S A241417 1,0,1,1,3,2,4,5,9,9,14,18,23,24,36,39,51,61,79,92,123,148,195,237,
%T A241417 297,359,464,552,679,822,1012,1183,1465,1707,2075,2438,2956,3433,4173,
%U A241417 4851,5837,6837,8218,9554,11518,13396,16022,18697,22300,25923,30873,35838
%N A241417 Number of partitions p of n such that the number of numbers p having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is not a part.
%F A241417 a(n) + A239737(n) = A000041(n) for n >= 0.
%e A241417 a(6) counts these 4 partitions: 6, 51, 33, 222.
%t A241417 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 A241417 Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}] (* A241413 *)
%t A241417 Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}] (* A241414 *)
%t A241417 Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
%t A241417 Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
%t A241417 Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
%t A241417 Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)
%Y A241417 Cf. A241413, A241414, A241415, A241416, A239737, A000041.
%K A241417 nonn,easy
%O A241417 0,5
%A A241417 _Clark Kimberling_, Apr 23 2014