%I #6 Mar 02 2020 18:54:38
%S 0,0,0,0,0,1,5,14,35,83,193,417,890,1847,3809,7805,15833,32028,64513,
%T 129671,260155,521775,1044982,2092692,4188168,8381434,16767650,
%U 33544423,67098683,134213022,268443023,536912014,1073846768,2147720476,4295440133,8590833907
%N Number of non-unimodal compositions of n covering an initial interval of positive integers.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A composition of n is a finite sequence of positive integers summing to n.
%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F For n > 0, a(n) = A107429(n) - A227038(n).
%e The a(5) = 1 through a(7) = 14 compositions:
%e (212) (213) (1213)
%e (312) (1312)
%e (1212) (2113)
%e (2112) (2122)
%e (2121) (2131)
%e (2212)
%e (3112)
%e (3121)
%e (11212)
%e (12112)
%e (12121)
%e (21112)
%e (21121)
%e (21211)
%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&!unimodQ[#]&]],{n,0,10}]
%Y Not requiring non-unimodality gives A107429.
%Y Not requiring the covering condition gives A115981.
%Y The complement is counted by A227038.
%Y A version for partitions is A332579, with complement A332577.
%Y Unimodal compositions are A001523.
%Y Non-unimodal permutations are A059204.
%Y Non-unimodal normal sequences are A328509.
%Y Numbers whose unsorted prime signature is not unimodal are A332282.
%Y Cf. A007052, A072704, A072706, A332281, A332284, A332287, A332578, A332639, A332642, A332669, A332834, A332870.
%K nonn
%O 0,7
%A _Gus Wiseman_, Mar 02 2020