%I #15 Jan 22 2024 13:01:57

%S 1,1,2,3,4,5,7,8,10,13,14,17,22,24,28,34,37,43,53,56,64,76,83,93,111,

%T 117,131,153,163,182,210,225,250,284,304,332,377,401,441,497,529,576,

%U 647,687,745,830,883,955,1062,1127,1216,1339,1422,1532,1684,1779,1914

%N Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.

%C First differs from A000041 at a(6) = 10, A000041(6) = 11.

%C A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%H Fausto A. C. Cariboni, <a href="/A332728/b332728.txt">Table of n, a(n) for n = 0..600</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</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(8) = 10 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (21) (22) (32) (33) (43) (44)

%e (111) (31) (41) (42) (52) (53)

%e (1111) (221) (51) (61) (62)

%e (11111) (222) (331) (71)

%e (321) (421) (332)

%e (111111) (2221) (431)

%e (1111111) (521)

%e (2222)

%e (11111111)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[IntegerPartitions[n],unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]

%Y The non-negated version is A332283.

%Y The non-negated complement is counted by A332284.

%Y The strict case is A332577.

%Y The case of run-lengths (instead of differences) is A332638.

%Y The complement is counted by A332744.

%Y The Heinz numbers of partitions not in this class are A332287.

%Y Unimodal compositions are A001523.

%Y Compositions whose negation is unimodal are A332578.

%Y Compositions whose run-lengths are unimodal are A332726.

%Y Cf. A007052, A332280, A332285, A332286, A332639, A332642, A332669, A332670, A332741.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 26 2020