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A332578 Number of compositions of n whose negation is unimodal. 43
1, 1, 2, 4, 7, 13, 21, 36, 57, 91, 140, 217, 323, 485, 711, 1039, 1494, 2144, 3032, 4279, 5970, 8299, 11438, 15708, 21403, 29065, 39218, 52725, 70497, 93941, 124562, 164639, 216664, 284240, 371456, 484004, 628419, 813669, 1050144, 1351757, 1734873, 2221018, 2835613 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A composition of n is a finite sequence of positive integers summing to n.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000 (terms 0..1000 from Andrew Howroyd)

FORMULA

a(n) + A332669(n) = 2^(n - 1).

G.f.: 1 + Sum_{j>0} x^j/((1 - x^j)*(Product_{k>j} 1 - x^k)^2). - Andrew Howroyd, Mar 01 2020

a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Mar 01 2020

EXAMPLE

The a(1) = 1 through a(5) = 13 compositions:

  (1)  (2)   (3)    (4)     (5)

       (11)  (12)   (13)    (14)

             (21)   (22)    (23)

             (111)  (31)    (32)

                    (112)   (41)

                    (211)   (113)

                    (1111)  (122)

                            (212)

                            (221)

                            (311)

                            (1112)

                            (2111)

                            (11111)

MATHEMATICA

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], unimodQ[-#]&]], {n, 0, 10}]

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 *)

PROG

(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

CROSSREFS

Dominated by A001523 (unimodal compositions).

The strict case is A072706.

The case that is unimodal also is A329398.

The complement is counted by A332669.

Row sums of A332670.

Unimodal normal sequences appear to be A007052.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Partitions whose run-lengths are unimodal are A332280.

Partitions whose negated run-lengths are unimodal are A332638.

Numbers whose unsorted prime signature is not unimodal are A332642.

Partitions whose negated 0-appended differences are unimodal are A332728.

Cf. A011782, A072704, A107429, A227038, A332282, A332283, A332639, A332741, A332742, A332744, A332832, A332870.

Sequence in context: A233759 A090752 A051058 * A026625 A026691 A018150

Adjacent sequences:  A332575 A332576 A332577 * A332579 A332580 A332581

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 28 2020

EXTENSIONS

Terms a(26) and beyond from Andrew Howroyd, Mar 01 2020

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)