A322600 a(n) is the number of unlabeled rank-3 graded lattices with 5 coatoms and n atoms.
1, 5, 20, 68, 190, 441, 907, 1690, 2916, 4734, 7310, 10836, 15528, 21619, 29365, 39045, 50961, 65434, 82809, 103453, 127751, 156117, 188980, 226794, 270037, 319204, 374813, 437409, 507553, 585831, 672847, 769233, 875637, 992735, 1121218, 1261802
Offset: 1
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).
Formula
For n>=3: a(n) = (175/192)n^4 - (3079/480)n^3 + (11771/480)n^2
- [7268/160, 7273/160]n
+ [33600, 34019, 34072, 33627, 33152, 34915, 33624, 33947, 33472, 33507,
34520, 34459, 32832, 33827, 34072, 34395, 33344, 34147, 33432, 33947,
34240, 33699, 33752, 34267, 32832, 34595, 34264, 33627, 33152, 34147,
34200, 34139, 33472, 33507, 33752, 35035, 33024, 33827, 34072, 33627,
33920, 34339, 33432, 33947, 33472, 34275, 33944, 34267, 32832, 33827,
34840, 33819, 33152, 34147, 33432, 34715, 33664, 33507, 33752, 34267] / 960.
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.
Conjectures from Colin Barker, Dec 20 2018: (Start)
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)).
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.
(End)