A279036 Number of partitions p of n that contain a proper partition of the maximal part of p.

0, 0, 0, 1, 1, 4, 4, 10, 12, 21, 25, 46, 50, 82, 99, 148, 175, 259, 303, 435, 513, 708, 845, 1146, 1347, 1802, 2138, 2793, 3318, 4273, 5050, 6471, 7621, 9641, 11406, 14210, 16758, 20833, 24475, 30143

Offset

1,6

Links

Table of n, a(n) for n=1..40.

Example

a(8) counts these 10 partitions:  [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,1,1], [2,2,1,1,1,1],[2,1,1,1,1,1,1]; e.g., [3,1] is a proper partition of 4.

Mathematica

Table[parts = IntegerPartitions[z]; parts = Drop[parts,
Position[Map[#[[1]] &, parts], Floor[z/2], 1, 1][[1]][[1]] - 1];
Count[Table[{first, rest} = {First[#], Rest[#]} &[parts[[nn]]];
Apply[Or, Map[MatchQ[rest, #] &, Map[Flatten[{___, #, ___}] &,
Rest[IntegerPartitions[first]]]]], {nn, Length[parts]}], True], {z, 30}]
(* Peter J. C. Moses, Dec 02 2016 *)

Crossrefs

Cf. A000041.

Sequence in context: A219471 A006477 A233739 * A182699 A058596 A180964

Adjacent sequences: A279033 A279034 A279035 * A279037 A279038 A279039

Keyword

nonn,easy

Author

Clark Kimberling, Dec 04 2016

Status

approved