A241515 Number of partitions of n such that (number parts having multiplicity 1) is a part or (number of parts > 1) is a part.

0, 1, 0, 1, 3, 5, 7, 12, 14, 21, 26, 37, 47, 69, 81, 114, 145, 194, 245, 329, 403, 537, 665, 853, 1055, 1358, 1649, 2096, 2551, 3182, 3887, 4816, 5800, 7174, 8646, 10536, 12680, 15434, 18398, 22272, 26578, 31900, 37949, 45433, 53751, 64125, 75774, 89800

Offset

0,5

Links

Table of n, a(n) for n=0..47.

Formula

a(n) + A241514(n) = A000041(n) for n >= 0.

Example

a(6) counts these 7 partitions:  51, 42, 411, 321, 3111, 2211, 21111.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==       1 &]]];
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, Length[p] - Count[p, 1]]], {n, 0, z}]  (* A241511 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241512 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241513 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241514 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, Length[p] - Count[p, 1]] ], {n, 0, z}] (* A241515 *)

Crossrefs

Cf. A241506, A241511, A241511, A241512, A241513, A241514.

Sequence in context: A243179 A207459 A368036 * A070334 A137700 A325267

Adjacent sequences: A241512 A241513 A241514 * A241516 A241517 A241518

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2014

Status

approved