A241444 Number of partitions of n such that the number of parts having multiplicity 1 is a part and the number of distinct parts is not a part.

0, 0, 0, 0, 0, 1, 2, 4, 5, 9, 12, 17, 21, 34, 38, 57, 72, 97, 121, 169, 204, 279, 338, 440, 551, 703, 857, 1095, 1341, 1664, 2038, 2536, 3061, 3777, 4578, 5564, 6726, 8170, 9776, 11849, 14131, 16992, 20246, 24264, 28703, 34322, 40535, 48156, 56761, 67239

Offset

0,7

Links

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

Formula

a(n) + A241442(n) + A241443(n) = A241446(n) for n >= 0.

Example

a(6) counts these 2 partitions:  411, 3111.

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. A241442, A241443, A241445, A241446.

Sequence in context: A236336 A239515 A087667 * A082592 A241339 A327781

Adjacent sequences: A241441 A241442 A241443 * A241445 A241446 A241447

Keyword

nonn,easy

Author

Clark Kimberling, Apr 23 2014

Status

approved