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%I #8 May 19 2014 15:08:39
%S 0,1,0,0,0,0,0,2,1,4,4,9,12,24,25,44,57,84,109,159,193,277,344,458,
%T 571,763,923,1211,1474,1874,2305,2902,3494,4399,5314,6543,7907,9733,
%U 11609,14198,16993,20539,24512,29557,35032,42082,49858,59373,70194,83490
%N Number of partitions of n such that (number parts having multiplicity 1) is a part and (number of parts > 1) is not a part.
%F a(n) + A241511(n) + A241512(n) = A241515(n) for n >= 0.
%e a(10) counts these 4 partitions: 631, 5221, 42211, 32221.
%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, A241512, A241514, A241515.
%K nonn,easy
%O 0,8
%A _Clark Kimberling_, Apr 24 2014
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