A241377 Number of partitions of n such that the number of parts is a part and the number of distinct parts is a part.

0, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 6, 8, 11, 12, 16, 20, 27, 31, 40, 49, 59, 72, 95, 110, 133, 164, 196, 237, 289, 351, 410, 502, 595, 704, 843, 1009, 1193, 1422, 1658, 1983, 2332, 2744, 3204, 3796, 4459, 5189, 6083, 7116, 8292, 9677, 11222, 13041, 15235

Offset

0,7

Links

Table of n, a(n) for n=0..53.

Formula

a(n) + A241378(n) + A241379(n) = A241381(n) for n >= 0.

Example

a(9) counts these 4 partitions:  72, 531, 432, 4311.

Mathematica

z = 70; 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. A241378, A241379, A241380, A241381.

Sequence in context: A063595 A316080 A130082 * A266750 A143065 A192660

Adjacent sequences: A241374 A241375 A241376 * A241378 A241379 A241380

Keyword

nonn,easy

Author

Clark Kimberling, Apr 21 2014

Status

approved