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%I #8 May 19 2014 15:07:39
%S 0,0,0,0,2,1,2,4,4,4,5,8,9,10,13,14,21,24,31,41,52,67,89,110,134,182,
%T 219,280,337,429,523,640,785,959,1168,1416,1714,2083,2482,2971,3596,
%U 4282,5103,6079,7236,8555,10176,11958,14129,16668,19545,22949,26939
%N Number of partitions of n such that (number parts having multiplicity 1) is not a part and (number of parts > 1) is a part.
%F a(n) + A241511(n) + A241513(n) = A241515(n) for n >= 0.
%e a(6) counts these 2 partitions: 51, 2211.
%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, Length[p] - Count[p, 1]]], {n, 0, z}] (* A241511 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241512 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241513 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241514 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241515 *)
%Y Cf. A241506, A241511, A241511, A241513, A241514, A241515.
%K nonn,easy
%O 0,5
%A _Clark Kimberling_, Apr 24 2014