A241379 Number of partitions of n such that the number of parts is a part and the number of distinct parts is not a part.

0, 0, 0, 0, 1, 1, 0, 2, 1, 3, 3, 6, 6, 10, 12, 18, 21, 28, 35, 48, 56, 78, 93, 115, 143, 187, 219, 282, 337, 419, 496, 629, 736, 912, 1090, 1324, 1564, 1901, 2238, 2720, 3187, 3821, 4501, 5387, 6291, 7455, 8770, 10341, 12080, 14227, 16575, 19479, 22676

Offset

0,8

Links

Table of n, a(n) for n=0..52.

Formula

a(n) + A241377(n) + A241378(n) = A241381(n) for n >= 0.

Example

a(9) counts these 3 partitions:  51111, 4221, 333.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := [p] = Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241377 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241378 *)
Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && ! MemberQ[p, d[p]]], {n, 0, z}]  (* A241379 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && ! MemberQ[p, d[p]]], {n, 0, z}]  (* A241380 *)
Table[Count[f[n], p_ /; MemberQ[p, Length[p]] || MemberQ[p, d[p]]], {n, 0, z}] (* A241381 *)

Crossrefs

Cf. A241377, A241378, A241380, A241381.

Sequence in context: A214920 A096373 A216961 * A108949 A167704 A109522

Adjacent sequences: A241376 A241377 A241378 * A241380 A241381 A241382

Keyword

nonn,easy

Author

Clark Kimberling, Apr 21 2014

Status

approved