A241416 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.

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, 312, 379, 480, 589, 730, 903, 1136, 1373, 1725, 2095, 2593, 3129, 3870, 4625, 5677, 6774, 8215, 9759, 11813, 13896, 16738, 19675, 23515, 27580, 32846, 38349, 45528

Offset: 0

Author

Clark Kimberling, Apr 23 2014

Keywords

• nonn
• easy

Examples

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

Crossrefs

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

Programs

Mathematica

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 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

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