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%I #9 Jan 05 2021 12:03:31
%S 1,1,2,2,4,5,6,8,12,13,19,24,26,33,45,52,66,78,92,113,129,160,192,231,
%T 268,305,361,436,501,591,665,783,897,1071,1228,1361,1593,1834,2101,
%U 2452,2685,3129,3526,4067,4568,5189,5868,6655,7565,8468,9400
%N Number of integer partitions 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 The case for all degrees including 1 is A325852.
%H Fausto A. C. Cariboni, <a href="/A325874/b325874.txt">Table of n, a(n) for n = 0..220</a>
%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(9) = 13 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (11) (21) (22) (32) (33) (43) (44) (54)
%e (31) (41) (42) (52) (53) (63)
%e (211) (221) (51) (61) (62) (72)
%e (311) (411) (322) (71) (81)
%e (2211) (331) (332) (441)
%e (421) (422) (522)
%e (511) (431) (621)
%e (521) (711)
%e (611) (4221)
%e (3221) (4311)
%e (3311) (5211)
%e (32211)
%t Table[Length[Select[IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,30}]
%Y Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325875, A325876.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jun 02 2019
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