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A322600 a(n) is the number of unlabeled rank-3 graded lattices with 5 coatoms and n atoms. 4

%I

%S 1,5,20,68,190,441,907,1690,2916,4734,7310,10836,15528,21619,29365,

%T 39045,50961,65434,82809,103453,127751,156117,188980,226794,270037,

%U 319204,374813,437409,507553,585831,672847,769233,875637,992735,1121218,1261802

%N a(n) is the number of unlabeled rank-3 graded lattices with 5 coatoms and n atoms.

%H Jukka Kohonen, <a href="/A322600/b322600.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 For n>=3: a(n) = (175/192)n^4 - (3079/480)n^3 + (11771/480)n^2

%F - [7268/160, 7273/160]n

%F + [33600, 34019, 34072, 33627, 33152, 34915, 33624, 33947, 33472, 33507,

%F 34520, 34459, 32832, 33827, 34072, 34395, 33344, 34147, 33432, 33947,

%F 34240, 33699, 33752, 34267, 32832, 34595, 34264, 33627, 33152, 34147,

%F 34200, 34139, 33472, 33507, 33752, 35035, 33024, 33827, 34072, 33627,

%F 33920, 34339, 33432, 33947, 33472, 34275, 33944, 34267, 32832, 33827,

%F 34840, 33819, 33152, 34147, 33432, 34715, 33664, 33507, 33752, 34267] / 960.

%F 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 60) is 0, 1, 2, ..., 59.

%F Conjectures from _Colin Barker_, Dec 20 2018: (Start)

%F G.f.: x*(1 + 4*x + 14*x^2 + 43*x^3 + 102*x^4 + 184*x^5 + 282*x^6 + 368*x^7 + 411*x^8 + 400*x^9 + 333*x^10 + 237*x^11 + 142*x^12 + 70*x^13 + 26*x^14 + 7*x^15 + x^16) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)).

%F a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-6) - a(n-7) + a(n-8) + a(n-9) + a(n-10) - a(n-13) - a(n-14) + a(n-15) for n>15.

%F (End)

%Y Fifth row of A300260.

%Y Previous rows are A322598, A322599.

%K nonn,easy

%O 1,2

%A _Jukka Kohonen_, Dec 19 2018

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