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%I #24 Mar 12 2018 04:58:55
%S 0,0,0,1,2,4,8,18,38,88,210,528,1396,3946,11896,38644,135790,518645,
%T 2160112,9832013,48945468,266458643
%N a(n) is the number of unlabeled, graded rank-3 lattices with n elements.
%C A graded lattice has rank 3 if its maximal chains have length 3.
%C They can be enumerated with a program such as that by Kohonen (2017).
%C Also called "two level lattices": apart from top and bottom, they have just coatoms and atoms. (Kleitman and Winston 1980)
%C Asymptotic upper bound: a(n) < b^(n^(3/2) + o(n^(3/2))), where b is about 1.699408. (Kleitman and Winston 1980)
%H D. J. Kleitman and K. J. Winston, <a href="http://dx.doi.org/10.1016/S0167-5060(08)70708-8">The asymptotic number of lattices</a>, Ann. Discrete Math. 6 (1980), 243-249.
%H J. Kohonen, <a href="http://arxiv.org/abs/1708.03750">Generating modular lattices up to 30 elements</a>, arXiv:1708.03750 [math.CO] preprint (2017).
%F a(n) = Sum_{k=1..n-3} A300260(n-2-k, k).
%e a(4)=1: The only possibility is the chain of length 3 (with 4 elements).
%e a(6)=4: These are the four lattices.
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%Y Cf. A278691 (unlabeled graded lattices).
%K nonn,more
%O 1,5
%A _Jukka Kohonen_, Mar 01 2018
%E a(22) from _Jukka Kohonen_, Mar 03 2018
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