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%I #6 Jan 07 2022 15:54:26
%S 0,0,0,0,0,0,0,1,2,5,8,15,23,37,52,80,109,156,208,289,378,509,654,865,
%T 1098,1425,1789,2290,2852,3603,4450,5569,6830,8467,10321,12701,15393,
%U 18805,22678,27535,33057,39908,47701,57304,68226,81572,96766,115212,136201
%N Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.
%C Also the number of non-weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence involves the somewhat degenerate case where no strict increases are allowed.
%F a(n) = A349061(n) - A347548(n).
%e The a(7) = 1 through a(11) = 15 partitions:
%e (3211) (4211) (3321) (5311) (4322)
%e (32111) (4311) (6211) (4421)
%e (5211) (32221) (5411)
%e (42111) (33211) (6311)
%e (321111) (43111) (7211)
%e (52111) (42221)
%e (421111) (43211)
%e (3211111) (53111)
%e (62111)
%e (322211)
%e (332111)
%e (431111)
%e (521111)
%e (4211111)
%e (32111111)
%t whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t Table[Length[Select[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]
%Y Counting all non-strict partitions gives A047967.
%Y Signatures of this type are counted by A274230, complement A027383.
%Y The strict instead of non-strict version is A347548, ranked by A350352.
%Y The version for compositions allowing strict is A349053, ranked by A349057.
%Y Allowing strict partitions gives A349061, complement A349060.
%Y The complement in non-strict partitions is A349795.
%Y These partitions are ranked by A350140 = A349794 \ A005117.
%Y A000041 = integer partitions, strict A000009.
%Y A001250 = alternating permutations, complement A348615.
%Y A003242 = Carlitz (anti-run) compositions.
%Y A025047 = alternating compositions, ranked by A345167.
%Y A025048/A025049 = directed alternating compositions.
%Y A096441 = weakly alternating 0-appended partitions.
%Y A345170 = partitions w/ an alternating permutation, ranked by A345172.
%Y A349052 = weakly alternating compositions.
%Y A349056 = weakly alternating permutations of prime indices.
%Y A349798 = weakly but not strongly alternating permutations of prime indices.
%Y Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.
%K nonn
%O 0,9
%A _Gus Wiseman_, Dec 25 2021