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%I #6 Mar 19 2023 15:31:40
%S 0,0,0,1,0,2,1,3,2,5,4,8,7,12,12,19,19,29,31,43,48,65,73,97,110,142,
%T 164,208,240,301,350,432,504,617,719,874,1019,1228,1434,1717,2001,
%U 2385,2778,3292,3831,4522,5252,6177,7164,8392,9722,11352,13125,15283,17643
%N Number of integer partitions of n whose first differences have mean -1.
%C These are partitions where the first part minus the last part is the number of parts minus 1.
%e The a(3) = 1 through a(11) = 8 partitions:
%e (21) . (32) (321) (43) (422) (54) (442) (65)
%e (311) (331) (4211) (432) (4321) (533)
%e (4111) (4221) (4411) (4331)
%e (4311) (52111) (4421)
%e (51111) (5222)
%e (52211)
%e (53111)
%e (611111)
%e For example, the partition y = (4,2,2,1) has first differences (-2,0,-1), with mean -1, so y is counted under a(9).
%t Table[Length[Select[IntegerPartitions[n],Mean[Differences[#]]==-1&]],{n,0,30}]
%Y For mean 0 we have A032741.
%Y The 0-appended version is A047993.
%Y For any negative mean we have A144300.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284/A058398/A327482 count partitions by mean.
%Y A067538 counts partitions with integer mean, ranks A316413.
%Y A326567/A326568 gives mean of prime indices, conjugate A326839/A326840.
%Y A360614/A360615 gives mean of 0-appended first differences of prime indices.
%Y Cf. A102627, A237363, A237832, A348551, A360688.
%K nonn
%O 0,6
%A _Gus Wiseman_, Mar 13 2023
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