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A332870 Number of compositions of n that are neither unimodal nor is their negation.

%I A332870 #12 Jan 01 2021 18:13:57
%S A332870 0,0,0,0,0,0,2,9,32,92,243,587,1361,3027,6564,13928,29127,60180,
%T A332870 123300,250945,508326,1025977,2065437,4150056,8327344,16692844,
%U A332870 33438984,66951671,134004892,268148573,536486146,1073227893,2146800237,4294061970,8588740071,17178298617
%N A332870 Number of compositions of n that are neither unimodal nor is their negation.
%C A332870 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332870 A composition of n is a finite sequence of positive integers summing to n.
%H A332870 Andrew Howroyd, <a href="/A332870/b332870.txt">Table of n, a(n) for n = 0..1000</a>
%F A332870 a(n) = 2^(n-1) - A001523(n) - A332578(n) + 2*A000041(n) - A000005(n) for n > 0. - _Andrew Howroyd_, Dec 30 2020
%e A332870 The a(6) = 2 and a(7) = 9 compositions:
%e A332870   (1212)  (1213)
%e A332870   (2121)  (1312)
%e A332870           (2131)
%e A332870           (3121)
%e A332870           (11212)
%e A332870           (12112)
%e A332870           (12121)
%e A332870           (21121)
%e A332870           (21211)
%t A332870 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332870 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[#]&&!unimodQ[-#]&]],{n,0,10}]
%Y A332870 The case of run-lengths of partitions is A332640.
%Y A332870 The version for unsorted prime signature is A332643.
%Y A332870 Unimodal compositions are A001523.
%Y A332870 Non-unimodal compositions are A115981.
%Y A332870 Non-unimodal normal sequences are A328509.
%Y A332870 Compositions whose negation is unimodal are A332578.
%Y A332870 Compositions whose negation is not unimodal are A332669.
%Y A332870 Partitions with weakly increasing or decreasing run-lengths are A332745.
%Y A332870 Compositions that are neither weakly increasing nor decreasing are A332834.
%Y A332870 Compositions with weakly increasing or decreasing run-lengths are A332835.
%Y A332870 Cf. A000005, A000041, A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332641, A332746, A332831, A332833.
%K A332870 nonn
%O A332870 0,7
%A A332870 _Gus Wiseman_, Mar 02 2020
%E A332870 Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2020