This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A332727 #15 Feb 16 2025 08:33:59
%S A332727 0,0,0,0,0,0,1,3,8,28,74,188,468,1120,2596,5944,13324,29437,64288,
%T A332727 138929,297442,632074,1333897,2798352,5840164,12132638,25102232,
%U A332727 51750419,106346704,217921161,445424102,908376235,1848753273,3755839591,7617835520,15428584567,31207263000
%N A332727 Number of compositions of n whose run-lengths are not unimodal.
%C A332727 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332727 A composition of n is a finite sequence of positive integers summing to n.
%H A332727 Andrew Howroyd, <a href="/A332727/b332727.txt">Table of n, a(n) for n = 0..500</a>
%H A332727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F A332727 a(n) + A332726(n) = 2^(n - 1).
%e A332727 The a(6) = 1 through a(8) = 8 compositions:
%e A332727 (11211) (11311) (11411)
%e A332727 (111211) (111311)
%e A332727 (112111) (112112)
%e A332727 (113111)
%e A332727 (211211)
%e A332727 (1111211)
%e A332727 (1112111)
%e A332727 (1121111)
%t A332727 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332727 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,10}]
%Y A332727 Looking at the composition itself (not its run-lengths) gives A115981.
%Y A332727 The case of partitions is A332281, with complement counted by A332280.
%Y A332727 The complement is counted by A332726.
%Y A332727 Unimodal compositions are A001523.
%Y A332727 Non-unimodal normal sequences are A328509.
%Y A332727 Compositions with normal run-lengths are A329766.
%Y A332727 Numbers whose prime signature is not unimodal are A332282.
%Y A332727 Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
%Y A332727 Compositions whose negation is not unimodal are A332669.
%Y A332727 Compositions whose run-lengths are weakly increasing are A332836.
%Y A332727 Cf. A007052, A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332578, A332638, A332639, A332670, A332741, A332833.
%K A332727 nonn
%O A332727 0,8
%A A332727 _Gus Wiseman_, Feb 29 2020
%E A332727 Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2020