%I #4 Apr 29 2014 22:45:31
%S 0,0,0,0,1,1,2,2,3,3,5,7,12,11,19,23,35,35,53,59,90,102,138,156,220,
%T 259,331,402,515,607,771,912,1169,1363,1699,2011,2513,2941,3603,4255,
%U 5230,6096,7438,8695,10546,12344,14797,17301,20760,24186,28783,33566
%N Number of partitions of n such that (number parts having multiplicity 1) is not a part and (number of 1s) is a part.
%F a(n) + A241506(n) + A241508(n) = A241510(n) for n >= 0.
%e a(6) counts these 3 partitions: 42, 411, 2111.
%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, A241508, A241509, A241510.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_, Apr 24 2014