%I #8 Feb 18 2020 04:47:59

%S 1,1,2,4,7,12,24,39,72,125,224,387,697,1205,2141,3736,6598,11516,

%T 20331,35526,62507,109436,192200,336533,590582,1034187

%N Number of alternately co-strong compositions of n.

%C A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

%e The a(1) = 1 through a(5) = 12 compositions:

%e (1) (2) (3) (4) (5)

%e (11) (12) (13) (14)

%e (21) (22) (23)

%e (111) (31) (32)

%e (112) (41)

%e (121) (113)

%e (1111) (131)

%e (212)

%e (221)

%e (1112)

%e (1121)

%e (11111)

%e For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).

%t tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tniQ]],{n,0,10}]

%Y The case of partitions is A317256.

%Y The recursive (rather than alternating) version is A332274.

%Y The total (rather than alternating) version is (also) A332274.

%Y The strong version is this same sequence.

%Y The case of reversed partitions is A332339.

%Y The normal version is A332340(n) + 1 for n > 1.

%Y Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Feb 17 2020