A308410 a(n) is the number of partitions p = p(1) >= p(2) >= ... >= p(k) of n whose alternating sum is a part of p.

1, 1, 3, 2, 5, 6, 10, 10, 20, 18, 33, 35, 55, 59, 92, 97, 146, 161, 231, 251, 363, 393, 551, 609, 828, 924, 1240, 1382, 1824, 2055, 2665, 3004, 3870, 4359, 5551, 6280, 7910, 8957, 11201, 12683, 15728, 17857, 21951, 24939, 30472, 34625, 42031, 47803, 57677

Offset

1,3

Links

Table of n, a(n) for n=1..49.

Example

The a(6) = 6 partitions of 6 to be counted are these:
[6] has alternating sum 6, which is a part,
[4,2] has alternating sum 4 - 2 = 2, a part,
[4,1,1] has alternating sum 4 - 1 + 1 = 4,
[3,2,1] has alternating sum 3 - 2 + 1 = 2,
[2,2,2] has alternating sum 2 - 2 + 1 = 2, and
[2,1,1,1,1] has alternating sum 2 - 1 + 1 - 1 + 1 - 1 = 2.

Mathematica

Map[Count[Map[Apply[MemberQ, {#, Total[Map[
Total, {Take[##], Drop[##]} &[#, {1, -1, 2}] {1, -1}]]}] &,
IntegerPartitions[#]], True] &, Range[40]]
(* Peter J. C. Moses, May 25 2019 *)

Crossrefs

Cf. A000041, A308230.

Sequence in context: A194012 A255556 A331462 * A286111 A338209 A240574

Adjacent sequences: A308407 A308408 A308409 * A308411 A308412 A308413

Keyword

nonn,easy

Author

Clark Kimberling, Jun 05 2019

Status

approved