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%I #5 Apr 28 2014 17:56:32
%S 0,1,0,1,1,3,3,4,5,8,9,12,17,25,30,43,52,73,93,119,147,191,238,303,
%T 370,473,573,721,873,1089,1326,1640,1954,2438,2900,3556,4240,5181,
%U 6140,7452,8851,10626,12600,15090,17812,21248,25063,29686,34969,41344,48465
%N 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.
%F a(n) + A241443(n) + A241444(n) = A241446(n) for n >= 0.
%e a(6) counts these 3 partitions: 42, 321, 21111.
%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241442 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241443 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241444 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241445 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, d[p]] ], {n, 0, z}] (* A241446 *)
%Y Cf. A241443, A241444, A241445, A241446.
%K nonn,easy
%O 0,6
%A _Clark Kimberling_, Apr 23 2014
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