A325875 Number of compositions of n whose differences of all degrees > 1 are nonzero.

1, 1, 2, 3, 7, 13, 20, 38, 69, 129, 222, 407, 726, 1313, 2318, 4146, 7432, 13296, 23759, 42458, 75714

Offset

0,3

Comments

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.

A composition of n is a finite sequence of positive integers with sum n.

The case for all degrees including 1 is A325851.

Links

Table of n, a(n) for n=0..20.

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

Example

The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1122)
                                  (1131)
                                  (1212)
                                  (1311)
                                  (2121)
                                  (2211)
                                  (11211)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, 2, Length[#]}], 0]&]], {n, 0, 10}]

Crossrefs

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325874, A325876.

Sequence in context: A156300 A067834 A070754 * A049887 A048216 A003509

Adjacent sequences: A325872 A325873 A325874 * A325876 A325877 A325878

Keyword

nonn,more

Author

Gus Wiseman, Jun 02 2019

Status

approved