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%I #5 Sep 21 2023 08:22:52
%S 1,1,1,1,1,1,1,2,2,4,4,7,7,12,12,20,20,30,31,45,46,66,68,93,97,130,
%T 136,179,188,242,256,325,344,432,459,568,606,742,793,963,1031,1240,
%U 1331,1589,1707,2026,2179,2567,2766,3240,3493,4072,4393,5094,5501,6351
%N Number of strict integer partitions of n that are not of length 2 and do not contain n/2.
%C Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.
%e The a(6) = 1 through a(12) = 7 strict partitions:
%e (6) (7) (8) (9) (10) (11) (12)
%e (4,2,1) (5,2,1) (4,3,2) (6,3,1) (5,4,2) (5,4,3)
%e (5,3,1) (7,2,1) (6,3,2) (7,3,2)
%e (6,2,1) (4,3,2,1) (6,4,1) (7,4,1)
%e (7,3,1) (8,3,1)
%e (8,2,1) (9,2,1)
%e (5,3,2,1) (5,4,2,1)
%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]
%Y The second condition alone has bisections A078408 and A365828.
%Y The complement is counted by A365659.
%Y The non-strict version is A365825, complement A238628.
%Y The first condition alone is A365827, complement A140106.
%Y A000041 counts integer partitions, strict A000009.
%Y A182616 counts partitions of 2n that do not contain n, strict A365828.
%Y Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.
%K nonn
%O 0,8
%A _Gus Wiseman_, Sep 20 2023