%I
%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 kth differences for k > 0 are the differences of the (k1)th differences. The differences of all degrees of a sequence are the union of its zeroth through mth 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
