%I #4 Apr 29 2014 00:06:53
%S 0,0,0,0,1,3,3,7,8,14,17,25,30,45,52,72,91,123,153,205,253,339,419,
%T 542,673,864,1051,1336,1625,2023,2461,3040,3642,4490,5383,6527,7837,
%U 9481,11291,13624,16208,19403,23087,27541,32619,38832,45923,54327,64150,75737
%N Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part and the number of numbers having multiplicity > 1 is a part.
%F a(n) + A241415(n) + A241416(n) = A239737(n) for n >= 0.
%e a(6) counts these 3 partitions: 411, 3111, 21111.
%t 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 Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}] (* A241413 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}] (* A241414 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]]  MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)
%Y Cf. A241413, A241415, A241416, A241417, A239737, A000041.
%K nonn,easy
%O 0,6
%A _Clark Kimberling_, Apr 23 2014
