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%I #13 Apr 17 2021 03:42:05
%S 0,0,0,0,0,0,0,0,0,0,10,10,20,30,50,150,180,290,420,630,860,1828,2168,
%T 3326,4514,6530,8576,12188,20096,25314,35576,48062,65592,86752,117222,
%U 152060,237590,292346,402798,524596,711270,910606,1221204,1554382,2044460,2927124
%N Number of strict compositions of n that are neither unimodal nor is their negation.
%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. It is strict if there are not repeated parts.
%H Andrew Howroyd, <a href="/A332874/b332874.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F G.f.: Sum_{k>=4} (k! - 2^k + 2) * [y^k](Product_{j>=1} 1 + y*x^j). - _Andrew Howroyd_, Apr 16 2021
%e The a(10) = 10 through a(12) = 20 compositions:
%e (1,3,2,4) (1,3,2,5) (1,3,2,6)
%e (1,4,2,3) (1,5,2,3) (1,4,2,5)
%e (2,1,4,3) (2,1,5,3) (1,5,2,4)
%e (2,3,1,4) (2,3,1,5) (1,6,2,3)
%e (2,4,1,3) (2,5,1,3) (2,1,5,4)
%e (3,1,4,2) (3,1,5,2) (2,1,6,3)
%e (3,2,4,1) (3,2,5,1) (2,3,1,6)
%e (3,4,1,2) (3,5,1,2) (2,4,1,5)
%e (4,1,3,2) (5,1,3,2) (2,5,1,4)
%e (4,2,3,1) (5,2,3,1) (2,6,1,3)
%e (3,1,6,2)
%e (3,2,6,1)
%e (3,6,1,2)
%e (4,1,5,2)
%e (4,2,5,1)
%e (4,5,1,2)
%e (5,1,4,2)
%e (5,2,4,1)
%e (6,1,3,2)
%e (6,2,3,1)
%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],UnsameQ@@#&&!unimodQ[#]&&!unimodQ[-#]&]],{n,0,20}]
%o (PARI) seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=4, n, (k! - 2^k + 2)*polcoef(p,k,y)), -(n+1))} \\ _Andrew Howroyd_, Apr 16 2021
%Y The non-strict version for unsorted prime signature is A332643.
%Y The non-strict version is A332870.
%Y Unimodal compositions are A001523.
%Y Non-unimodal compositions are A115981.
%Y Non-unimodal normal sequences are A328509.
%Y Compositions whose negation is unimodal are A332578.
%Y Compositions whose negation is not unimodal are A332669.
%Y Compositions with neither weakly increasing nor weakly decreasing run-lengths are A332833.
%Y Compositions with weakly increasing or weakly decreasing run-lengths are A332835.
%Y Cf. A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332640, A332641, A332745, A332746, A332831, A332834.
%K nonn
%O 0,11
%A _Gus Wiseman_, Mar 04 2020
%E Terms a(21) and beyond from _Andrew Howroyd_, Apr 16 2021
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