A277753 Number of partitions p of n that contain a proper partition of the number of parts of p.

0, 0, 0, 0, 1, 2, 4, 8, 12, 19, 28, 42, 58, 82, 110, 150, 200, 263, 344, 450, 578, 741, 947, 1197, 1513, 1899, 2374, 2954, 3669, 4529, 5576, 6849, 8380, 10223, 12449

Offset

1,6

Links

Table of n, a(n) for n=1..35.

Example

a(9) counts these 12 partitions: [6,2,1], [5,2,1,1], [4,3,1,1], [4,2,2,1], [4,2,1,1,1], [3,3,1,1,1], [3,2,2,2], [3,2,2,1,1], [3,2,1,1,1,1], [2,2,2,2,1], [2,2,2,1,1,1],[2,2,1,1,1,1,1]; e.g., [2,2,1] is a proper partition of 5, which is the number of parts in [3,2,2,1,1].

Mathematica

Table[parts = IntegerPartitions[z];
Count[Table[rest = Rest[parts[[nn]]];
seq = Map[{#, Flatten[{___, #, ___}]} &,
Rest[IntegerPartitions[Length[rest] + 1]]];
Apply[Or, Map[MatchQ[rest, #[[2]]] &, seq]], {nn, Length[parts]}],
True], {z, 30}]  (* Peter J. C. Moses, Dec 06 2016 *)

Crossrefs

Cf. A000041.

Sequence in context: A136184 A011908 A117455 * A237820 A110571 A037168

Adjacent sequences: A277750 A277751 A277752 * A277754 A277755 A277756

Keyword

nonn,easy

Author

Clark Kimberling, Dec 07 2016

Status

approved