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%I #5 Jun 02 2019 23:40:14
%S 1,1,2,3,7,13,20,38,69,129,222,407,726,1313,2318,4146,7432,13296,
%T 23759,42458,75714
%N Number of compositions of n whose differences of all degrees > 1 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. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.
%C A composition of n is a finite sequence of positive integers with sum n.
%C The case for all degrees including 1 is A325851.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>
%e The a(1) = 1 through a(6) = 20 compositions:
%e (1) (2) (3) (4) (5) (6)
%e (11) (12) (13) (14) (15)
%e (21) (22) (23) (24)
%e (31) (32) (33)
%e (112) (41) (42)
%e (121) (113) (51)
%e (211) (122) (114)
%e (131) (132)
%e (212) (141)
%e (221) (213)
%e (311) (231)
%e (1121) (312)
%e (1211) (411)
%e (1122)
%e (1131)
%e (1212)
%e (1311)
%e (2121)
%e (2211)
%e (11211)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,10}]
%Y Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325874, A325876.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jun 02 2019
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