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A300221 a(n) is the number of unlabeled, graded rank-3 lattices with n elements.

%I A300221 #24 Mar 12 2018 04:58:55
%S A300221 0,0,0,1,2,4,8,18,38,88,210,528,1396,3946,11896,38644,135790,518645,
%T A300221 2160112,9832013,48945468,266458643
%N A300221 a(n) is the number of unlabeled, graded rank-3 lattices with n elements.
%C A300221 A graded lattice has rank 3 if its maximal chains have length 3.
%C A300221 They can be enumerated with a program such as that by Kohonen (2017).
%C A300221 Also called "two level lattices": apart from top and bottom, they have just coatoms and atoms. (Kleitman and Winston 1980)
%C A300221 Asymptotic upper bound: a(n) < b^(n^(3/2) + o(n^(3/2))), where b is about 1.699408. (Kleitman and Winston 1980)
%H A300221 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 A300221 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 A300221 a(n) = Sum_{k=1..n-3} A300260(n-2-k, k).
%e A300221 a(4)=1: The only possibility is the chain of length 3 (with 4 elements).
%e A300221 a(6)=4: These are the four lattices.
%e A300221     o       o      o       o
%e A300221     |      / \    / \     /|\
%e A300221     o      o o    o o    o o o
%e A300221    /|\     | |    |/|     \|/
%e A300221   o o o    o o    o o      o
%e A300221    \|/     \ /    \ /      |
%e A300221     o       o      o       o
%Y A300221 Cf. A278691 (unlabeled graded lattices).
%K A300221 nonn,more
%O A300221 1,5
%A A300221 _Jukka Kohonen_, Mar 01 2018
%E A300221 a(22) from _Jukka Kohonen_, Mar 03 2018