A241417 Number of partitions p of n such that the number of numbers p having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is not a part.

1, 0, 1, 1, 3, 2, 4, 5, 9, 9, 14, 18, 23, 24, 36, 39, 51, 61, 79, 92, 123, 148, 195, 237, 297, 359, 464, 552, 679, 822, 1012, 1183, 1465, 1707, 2075, 2438, 2956, 3433, 4173, 4851, 5837, 6837, 8218, 9554, 11518, 13396, 16022, 18697, 22300, 25923, 30873, 35838

Offset

0,5

Links

Table of n, a(n) for n=0..51.

Formula

a(n) + A239737(n) = A000041(n) for n >= 0.

Example

a(6) counts these 4 partitions:  6, 51, 33, 222.

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 *)

Crossrefs

Cf. A241413, A241414, A241415, A241416, A239737, A000041.

Sequence in context: A275901 A305369 A097092 * A211363 A376901 A059320

Adjacent sequences: A241414 A241415 A241416 * A241418 A241419 A241420

Keyword

nonn,easy

Author

Clark Kimberling, Apr 23 2014

Status

approved