A241442 Number of partitions of n such that the number of parts having multiplicity 1 is a part and the number of distinct parts is a part.

0, 1, 0, 1, 1, 3, 3, 4, 5, 8, 9, 12, 17, 25, 30, 43, 52, 73, 93, 119, 147, 191, 238, 303, 370, 473, 573, 721, 873, 1089, 1326, 1640, 1954, 2438, 2900, 3556, 4240, 5181, 6140, 7452, 8851, 10626, 12600, 15090, 17812, 21248, 25063, 29686, 34969, 41344, 48465

Offset

0,6

Links

Table of n, a(n) for n=0..50.

Formula

a(n) + A241443(n) + A241444(n) = A241446(n) for n >= 0.

Example

a(6) counts these 3 partitions:  42, 321, 21111.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241442 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, d[p]] ], {n, 0,  z}] (* A241443 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241444 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241445 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, d[p]] ], {n, 0, z}] (* A241446 *)

Crossrefs

Cf. A241443, A241444, A241445, A241446.

Sequence in context: A100091 A239483 A104806 * A363091 A373721 A043551

Adjacent sequences: A241439 A241440 A241441 * A241443 A241444 A241445

Keyword

nonn,easy

Author

Clark Kimberling, Apr 23 2014

Status

approved