login
A070211
Number of compositions (ordered partitions) of n that are concave-down sequences.
19
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
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
KEYWORD
nice,nonn
AUTHOR
EXTENSIONS
Name edited by Gus Wiseman, May 15 2019
STATUS
approved