A241416 - OEIS

%I #4 Apr 29 2014 00:07:13

%S 0,1,0,1,0,1,2,1,2,3,4,4,8,14,16,28,33,47,61,83,98,131,157,201,248,

%T 312,379,480,589,730,903,1136,1373,1725,2095,2593,3129,3870,4625,5677,

%U 6774,8215,9759,11813,13896,16738,19675,23515,27580,32846,38349,45528

%N Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part and the number of numbers having multiplicity > 1 is not a part.

%F a(n) + A241414(n) + A241415(n) = A239737(n) for n >= 0.

%e a(6) counts these 2 partitions:  42, 321.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

%Y Cf. A241413, A241414, A241415, A241417, A239737.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 23 2014