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
Examples
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.
Programs
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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 *)