A070211 Number of compositions (ordered partitions) of n that are concave-down sequences.

1, 1, 2, 4, 6, 9, 14, 18, 24, 34, 42, 52, 68, 82, 101, 126, 147, 175, 213, 246, 289, 344, 392, 453, 530, 598, 687, 791, 885, 1007, 1151, 1276, 1438, 1629, 1806, 2018, 2262, 2490, 2775, 3091, 3387, 3754, 4165, 4542, 5011, 5527, 6012, 6600, 7245, 7864, 8614

Offset

0,3

Comments

Here, a finite sequence is concave if each term (other than the first or last) is at least the average of the two adjacent terms. - Eric M. Schmidt, Sep 29 2013

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1). Then a(n) is the number of compositions of n with weakly decreasing differences. - Gus Wiseman, May 15 2019

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

Out of the 8 ordered partitions of 4, only 2+1+1 and 1+1+2 are not concave, so a(4)=6.
From Gus Wiseman, May 15 2019: (Start)
The a(1) = 1 through a(6) = 14 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (122)    (51)
                            (131)    (123)
                            (221)    (132)
                            (11111)  (141)
                                     (222)
                                     (231)
                                     (321)
                                     (1221)
                                     (111111)
(End)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], GreaterEqual@@Differences[#]&]], {n, 0, 15}] (* Gus Wiseman, May 15 2019 *)

Prog

(Sage) def A070211(n) : return sum(all(2*p[i] >= p[i-1] + p[i+1] for i in range(1, len(p)-1)) for p in Compositions(n)) # Eric M. Schmidt, Sep 29 2013

Crossrefs

Cf. A000079, A001523 (weakly unimodal compositions), A069916, A175342, A320466, A325361 (concave-down partitions), A325545, A325546 (concave-up compositions), A325547, A325548, A325557.

Sequence in context: A198201 A281989 A328423 * A113753 A024457 A117842

Adjacent sequences: A070208 A070209 A070210 * A070212 A070213 A070214

Keyword

nice,nonn

Author

Pontus von Brömssen, May 07 2002

Extensions

Name edited by Gus Wiseman, May 15 2019

Status

approved