A241414 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part and the number of numbers having multiplicity > 1 is a part.

0, 0, 0, 0, 1, 3, 3, 7, 8, 14, 17, 25, 30, 45, 52, 72, 91, 123, 153, 205, 253, 339, 419, 542, 673, 864, 1051, 1336, 1625, 2023, 2461, 3040, 3642, 4490, 5383, 6527, 7837, 9481, 11291, 13624, 16208, 19403, 23087, 27541, 32619, 38832, 45923, 54327, 64150, 75737

Offset

0,6

Links

Table of n, a(n) for n=0..49.

Formula

a(n) + A241415(n) + A241416(n) = A239737(n) for n >= 0.

Example

a(6) counts these 3 partitions:  411, 3111, 21111.

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, A241415, A241416, A241417, A239737, A000041.

Sequence in context: A241637 A241641 A339398 * A218568 A218569 A218570

Adjacent sequences: A241411 A241412 A241413 * A241415 A241416 A241417

Keyword

nonn,easy

Author

Clark Kimberling, Apr 23 2014

Status

approved