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A332578 Number of compositions of n whose negation is unimodal.

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