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

1, 4, 13, 34, 68, 121, 197, 299, 432, 600, 806, 1055, 1352, 1698, 2100, 2561, 3085, 3675, 4338, 5074, 5891, 6790, 7777, 8854, 10029, 11300, 12677, 14160, 15756, 17465, 19297, 21249, 23332, 25544, 27894, 30381, 33016, 35794, 38728, 41815, 45065

Offset: 1

Author

Jukka Kohonen, Dec 19 2018

Keywords

• nonn
• easy

Examples

			a(2)=4: These are the four lattices.
    __o__        __o__       __o__       __o__
   / / \ \      / / \ \     / / \ \     / / \ \
  o o   o o     o o o o     o o o o     o o o o
   \_\ /_/|      \|/ \|      \|/  |     |/   \|
      o   o       o   o       o   o     o     o
       \ /         \ /         \ /       \_ _/
        o           o           o          o
		

Links

• Jukka Kohonen, Table of n, a(n) for n = 1..1000
• J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).

Crossrefs

Fourth row of A300260.

Adjacent rows are A322598, A322600.

Formula

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.

Conjectures from Colin Barker, Dec 19 2018: (Start)

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)).

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>10.

(End)