A360682 - OEIS

A360682

Number of integer partitions of n of length > 2 whose second differences have median 0.

%I #6 Feb 20 2023 21:49:20

%S 0,0,0,1,1,1,5,4,10,13,18,23,44,44,72,98,132,162,241,277,394,497,643,

%T 800,1076,1287,1660,2078,2604,3192,4065,4892,6113,7490,9166,11110,

%U 13717,16429,20033,24201,29143,34945,42251,50219,60253,71852,85503,101501,120899

%N Number of integer partitions of n of length > 2 whose second differences have median 0.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e The a(3) = 1 through a(9) = 13 partitions:

%e (111) (1111) (11111) (222) (22111) (2222) (333)

%e (321) (31111) (3221) (432)

%e (2211) (211111) (3311) (531)

%e (21111) (1111111) (22211) (22221)

%e (111111) (32111) (33111)

%e (41111) (51111)

%e (221111) (222111)

%e (311111) (321111)

%e (2111111) (411111)

%e (11111111) (2211111)

%e (3111111)

%e (21111111)

%e (111111111)

%t Table[Length[Select[IntegerPartitions[n],Median[Differences[#,2]]==0&]],{n,0,30}]

%Y For first differences we have A237363.

%Y For sum instead of median we have A360683.

%Y For mean instead of median we have A360683 - A008619.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

%Y A325347 counts partitions with integer median, strict A359907.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y A360005 gives median of prime indices (times two).

%Y Cf. A000975, A027193, A067538, A114638, A307683, A359908, A360245, A360254, A360687, A360688.

%K nonn

%O 0,7

%A _Gus Wiseman_, Feb 19 2023