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Number of integer partitions of n whose second differences sum to 0, meaning either there is only one part, or the first two parts have the same difference as the last two parts.
1

%I #6 Feb 20 2023 21:49:25

%S 1,1,2,3,4,4,8,6,11,12,17,14,32,23,40,44,64,59,104,93,149,157,218,227,

%T 342,349,481,538,713,777,1052,1145,1494,1692,2130,2416,3064,3449,4286,

%U 4918,6028,6882,8424,9620,11634,13396,16022,18416,22019,25248,29954

%N Number of integer partitions of n whose second differences sum to 0, meaning either there is only one part, or the first two parts have the same difference as the last two parts.

%e The a(1) = 1 through a(8) = 11 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) (11111) (51) (61) (62)

%e (222) (22111) (71)

%e (321) (1111111) (2222)

%e (2211) (3221)

%e (111111) (3311)

%e (22211)

%e (221111)

%e (11111111)

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

%Y For mean instead of sum we have a(n) - A008619(n).

%Y For median instead of sum we have A360682.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

%Y A067538 counts partitions with integer mean, strict A102627.

%Y A316413 ranks partitions with integer mean, complement A348551.

%Y Cf. A114638, A240219, A325347, A360068.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 19 2023