A241506 Number of partitions of n such that (number parts having multiplicity 1) is a part and (number of 1s) is a part.

0, 1, 0, 1, 1, 1, 2, 3, 5, 7, 11, 12, 17, 25, 32, 40, 54, 73, 95, 123, 152, 195, 252, 319, 395, 491, 624, 759, 951, 1167, 1446, 1767, 2147, 2631, 3212, 3881, 4684, 5672, 6848, 8215, 9825, 11809, 14070, 16818, 19957, 23737, 28169, 33377, 39357, 46546, 54814

Offset

0,7

Links

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

Formula

a(n) + A241507(n) + A241508(n) = A241510(n) for n >= 0.

Example

a(6) counts these 2 partitions:  51, 2211.

Mathematica

z = 52; 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, Count[p, 1]]], {n, 0, z}]  (* A241506 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241507 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241508 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241509 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, Count[p, 1]] ], {n, 0, z}] (* A241510 *)

Crossrefs

Cf. A241507, A241508, A241509, A241510.

Sequence in context: A139749 A178357 A205667 * A361851 A372055 A165132

Adjacent sequences: A241503 A241504 A241505 * A241507 A241508 A241509

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2014

Status

approved