%I #4 Apr 29 2014 22:45:48
%S 1,0,2,2,3,2,4,5,9,10,16,20,27,31,48,53,72,92,118,143,186,220,288,356,
%T 434,523,675,792,989,1205,1469,1754,2165,2565,3133,3752,4498,5345,
%U 6496,7629,9126,10869,12890,15212,18114,21220,25163,29611,34783,40756,48058
%N Number of partitions of n such that (number parts having multiplicity 1) is not a part and (number of 1s) is not a part.
%F a(n) + A241510(n) = A000041(n) for n >= 0.
%e a(6) counts these 4 partitions: 6, 33, 222, 111111.
%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. A241506, A241507, A241508, A241510, A000041.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Apr 24 2014