A241445 Number of partitions of n such that the number of parts having multiplicity 1 is not a part and the number of distinct parts is not a part.

1, 0, 1, 1, 3, 2, 4, 4, 7, 7, 11, 15, 21, 24, 36, 45, 59, 73, 99, 114, 155, 183, 241, 287, 371, 433, 567, 668, 842, 1003, 1270, 1483, 1856, 2205, 2707, 3210, 3940, 4627, 5661, 6656, 8050, 9489, 11432, 13385, 16070, 18855, 22459, 26310, 31253, 36487, 43249

Offset

0,5

Links

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

Formula

a(n) + A241446(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, #] == 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, A241444, A241446, A000041.

Sequence in context: A284013 A367219 A241412 * A371180 A147604 A095401

Adjacent sequences: A241442 A241443 A241444 * A241446 A241447 A241448

Keyword

nonn,easy

Author

Clark Kimberling, Apr 23 2014

Status

approved