%I #5 Apr 29 2014 22:45:23
%S 0,1,0,1,1,1,2,3,5,7,11,12,17,25,32,40,54,73,95,123,152,195,252,319,
%T 395,491,624,759,951,1167,1446,1767,2147,2631,3212,3881,4684,5672,
%U 6848,8215,9825,11809,14070,16818,19957,23737,28169,33377,39357,46546,54814
%N Number of partitions of n such that (number parts having multiplicity 1) is a part and (number of 1s) is a part.
%F a(n) + A241507(n) + A241508(n) = A241510(n) for n >= 0.
%e a(6) counts these 2 partitions: 51, 2211.
%t z = 52; 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, Count[p, 1]]], {n, 0, z}] (* A241506 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241507 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241508 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241509 *)
%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241510 *)
%Y Cf. A241507, A241508, A241509, A241510.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_, Apr 24 2014