A241411 Number of partitions of n such that the number of parts having multiplicity >1 is a part and the number of distinct parts is not a part.

0, 0, 0, 0, 0, 1, 2, 4, 5, 9, 12, 18, 23, 37, 44, 64, 80, 111, 139, 185, 235, 306, 380, 488, 611, 771, 956, 1191, 1472, 1823, 2238, 2748, 3345, 4098, 4967, 6025, 7279, 8797, 10558, 12709, 15204, 18215, 21692, 25880, 30702, 36545, 43194, 51166, 60314, 71255

Offset: 0

Author

Clark Kimberling, Apr 22 2014

Keywords

• nonn
• easy

Comments

As used here, the term "distinct parts" includes each number, once, that occurs more than once; e.g., the distinct parts of the partition {4,3,3,1,1,1} are 4, 3, 1.

Examples

			a(6) counts these 2 partitions:  411, 3111.
		

Crossrefs

Cf. A241408, A241409, A241410, A241412.

Programs

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)