A241412 Number of partitions of n such that neither the number of parts having multiplicity >1 nor the number of distinct parts is a part.

1, 0, 1, 1, 3, 2, 4, 4, 7, 7, 11, 14, 19, 21, 30, 38, 51, 59, 81, 98, 124, 156, 199, 239, 311, 365, 468, 572, 711, 844, 1070, 1271, 1572, 1884, 2318, 2749, 3387, 4000, 4879, 5796, 6977, 8266, 9986, 11769, 14071, 16632, 19800, 23300, 27700, 32471, 38447

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 4 partitions:  6, 51, 33, 222.
		

Crossrefs

Cf. A241408, A241409, A241410, A241411.

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  *)