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%I #18 Mar 01 2020 05:40:27
%S 1,1,2,4,7,13,21,36,57,91,140,217,323,485,711,1039,1494,2144,3032,
%T 4279,5970,8299,11438,15708,21403,29065,39218,52725,70497,93941,
%U 124562,164639,216664,284240,371456,484004,628419,813669,1050144,1351757,1734873,2221018,2835613
%N Number of compositions of n whose negation is unimodal.
%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 Vaclav Kotesovec, <a href="/A332578/b332578.txt">Table of n, a(n) for n = 0..3000</a> (terms 0..1000 from Andrew Howroyd)
%F a(n) + A332669(n) = 2^(n - 1).
%F G.f.: 1 + Sum_{j>0} x^j/((1 - x^j)*(Product_{k>j} 1 - x^k)^2). - _Andrew Howroyd_, Mar 01 2020
%F a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - _Vaclav Kotesovec_, Mar 01 2020
%e The a(1) = 1 through a(5) = 13 compositions:
%e (1) (2) (3) (4) (5)
%e (11) (12) (13) (14)
%e (21) (22) (23)
%e (111) (31) (32)
%e (112) (41)
%e (211) (113)
%e (1111) (122)
%e (212)
%e (221)
%e (311)
%e (1112)
%e (2111)
%e (11111)
%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],unimodQ[-#]&]],{n,0,10}]
%t nmax = 50; CoefficientList[Series[1 + Sum[x^j*(1 - x^j)/Product[1 - x^k, {k, j, nmax - j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 01 2020 *)
%o (PARI) seq(n)={Vec(1 + sum(j=1, n, x^j/((1-x^j)*prod(k=j+1, n-j, 1 - x^k + O(x*x^(n-j)))^2)))} \\ _Andrew Howroyd_, Mar 01 2020
%Y Dominated by A001523 (unimodal compositions).
%Y The strict case is A072706.
%Y The case that is unimodal also is A329398.
%Y The complement is counted by A332669.
%Y Row sums of A332670.
%Y Unimodal normal sequences appear to be A007052.
%Y Non-unimodal compositions are A115981.
%Y Non-unimodal normal sequences are A328509.
%Y Partitions whose run-lengths are unimodal are A332280.
%Y Partitions whose negated run-lengths are unimodal are A332638.
%Y Numbers whose unsorted prime signature is not unimodal are A332642.
%Y Partitions whose negated 0-appended differences are unimodal are A332728.
%Y Cf. A011782, A072704, A107429, A227038, A332282, A332283, A332639, A332741, A332742, A332744, A332832, A332870.
%K nonn
%O 0,3
%A _Gus Wiseman_, Feb 28 2020
%E Terms a(26) and beyond from _Andrew Howroyd_, Mar 01 2020
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