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A322599 a(n) is the number of unlabeled rank-3 graded lattices with 4 coatoms and n atoms.

%I A322599 #15 Dec 22 2018 03:56:44
%S A322599 1,4,13,34,68,121,197,299,432,600,806,1055,1352,1698,2100,2561,3085,
%T A322599 3675,4338,5074,5891,6790,7777,8854,10029,11300,12677,14160,15756,
%U A322599 17465,19297,21249,23332,25544,27894,30381,33016,35794,38728,41815,45065
%N A322599 a(n) is the number of unlabeled rank-3 graded lattices with 4 coatoms and n atoms.
%H A322599 Jukka Kohonen, <a href="/A322599/b322599.txt">Table of n, a(n) for n = 1..1000</a>
%H A322599 J. Kohonen, <a href="http://arxiv.org/abs/1804.03679">Counting graded lattices of rank three that have few coatoms</a>, arXiv:1804.03679 [math.CO] preprint (2018).
%F A322599 a(n) = (97/144)n^3 - (5/6)n^2 + [44/48, 47/48]n + [0, 13, 8, -45, 40, -19, 0, -5, 8, -27, 40, -37]/72. The value of the first bracket depends on whether n is even or odd. The value of the second bracket depends on whether (n mod 12) is 0, 1, 2, ..., 11.
%F A322599 Conjectures from _Colin Barker_, Dec 19 2018: (Start)
%F A322599 G.f.: x*(1 + 3*x + 8*x^2 + 17*x^3 + 21*x^4 + 21*x^5 + 16*x^6 + 7*x^7 + 3*x^8) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
%F A322599 a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>10.
%F A322599 (End)
%e A322599 a(2)=4: These are the four lattices.
%e A322599     __o__        __o__       __o__       __o__
%e A322599    / / \ \      / / \ \     / / \ \     / / \ \
%e A322599   o o   o o     o o o o     o o o o     o o o o
%e A322599    \_\ /_/|      \|/ \|      \|/  |     |/   \|
%e A322599       o   o       o   o       o   o     o     o
%e A322599        \ /         \ /         \ /       \_ _/
%e A322599         o           o           o          o
%Y A322599 Fourth row of A300260.
%Y A322599 Adjacent rows are A322598, A322600.
%K A322599 nonn,easy
%O A322599 1,2
%A A322599 _Jukka Kohonen_, Dec 19 2018