%I #4 Jun 02 2019 00:48:58

%S 1,1,1,3,4,7,12,21,35,58,102,171,284,485,819,1355,2301,3884,6528,

%T 10983,18380,30824,51851

%N Number of (strict) compositions of n whose differences of all degrees are nonzero.

%C 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. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

%e The a(1) = 1 through a(7) = 21 compositions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (12) (13) (14) (15) (16)

%e (21) (31) (23) (24) (25)

%e (121) (32) (42) (34)

%e (41) (51) (43)

%e (131) (132) (52)

%e (212) (141) (61)

%e (213) (124)

%e (231) (142)

%e (312) (151)

%e (1212) (214)

%e (2121) (232)

%e (241)

%e (313)

%e (412)

%e (421)

%e (1213)

%e (1312)

%e (2131)

%e (3121)

%e (12121)

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

%Y The case for only degrees > 1 is A325875.

%Y Cf. A049988, A175342, A238423, A295370, A325328, A325545, A325850, A325852, A325874.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, May 31 2019