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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.

%I A308410 #10 Jun 06 2019 02:38:45
%S A308410 1,1,3,2,5,6,10,10,20,18,33,35,55,59,92,97,146,161,231,251,363,393,
%T A308410 551,609,828,924,1240,1382,1824,2055,2665,3004,3870,4359,5551,6280,
%U A308410 7910,8957,11201,12683,15728,17857,21951,24939,30472,34625,42031,47803,57677
%N 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.
%e A308410 The a(6) = 6 partitions of 6 to be counted are these:
%e A308410 [6] has alternating sum 6, which is a part,
%e A308410 [4,2] has alternating sum 4 - 2 = 2, a part,
%e A308410 [4,1,1] has alternating sum 4 - 1 + 1 = 4,
%e A308410 [3,2,1] has alternating sum 3 - 2 + 1 = 2,
%e A308410 [2,2,2] has alternating sum 2 - 2 + 1 = 2, and
%e A308410 [2,1,1,1,1] has alternating sum 2 - 1 + 1 - 1 + 1 - 1 = 2.
%t A308410 Map[Count[Map[Apply[MemberQ, {#, Total[Map[
%t A308410 Total, {Take[##], Drop[##]} &[#, {1, -1, 2}] {1, -1}]]}] &,
%t A308410 IntegerPartitions[#]], True] &, Range[40]]
%t A308410 (* _Peter J. C. Moses_, May 25 2019 *)
%Y A308410 Cf. A000041, A308230.
%K A308410 nonn,easy
%O A308410 1,3
%A A308410 _Clark Kimberling_, Jun 05 2019