A241381 Number of partitions of n such that the number of parts or the number of distinct parts is a part.

0, 1, 1, 2, 3, 5, 5, 9, 11, 17, 22, 30, 41, 53, 73, 92, 121, 155, 200, 255, 324, 408, 516, 643, 796, 1009, 1231, 1529, 1872, 2317, 2792, 3452, 4168, 5073, 6115, 7433, 8875, 10741, 12816, 15400, 18344, 21923, 25997, 30999, 36693, 43412, 51334, 60629, 71339

Offset

0,4

Links

Table of n, a(n) for n=0..48.

Formula

a(n) + A241380(n) = A000041(n) for n >= 0.

Example

a(6) counts these 5 partitions:  41, 321, 2211, 21111, 111111.

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, A241379, A241380, A000041.

Sequence in context: A317081 A183562 A222705 * A237365 A322770 A257008

Adjacent sequences: A241378 A241379 A241380 * A241382 A241383 A241384

Keyword

nonn,easy

Author

Clark Kimberling, Apr 21 2014

Status

approved