OFFSET
1,1
COMMENTS
Conjecture: These are the number of linear intervals in the Cambrian lattices of type B_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term 34254 for n = 7.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1658
Clément Chenevière, Enumerative study of intervals in lattices of Tamari type, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 151.
FORMULA
From Peter Luschny, May 12 2021: (Start)
a(n) = 3*(2*n^3 + n - 1)*2^(2*n - 2)*binomial(n - 3/2, -1/2)/((n + 1)*n).
a(n) = [x^n] (15*x - 24*x^2 + 8*x^3 - 2 + (1 - 4*x)^(3/2)*(2 - 3*x))/(2*(1 - 4*x)^(3/2)*x).
a(n) ~ 4^(n-2)*(24*n - 15)/sqrt(Pi*n). (End)
a(n) = a(n-1)*2*(2*n - 3)*(2*n^3 + n - 1)/((n + 1)*(2*n^3 - 6*n^2 + 7*n - 4)) for n > 1. - Chai Wah Wu, May 13 2021
EXAMPLE
For B_2, among the 18 intervals in the hexagon-shaped lattice, only one is not linear.
MAPLE
a := n -> 3*(2*n^3 + n - 1)*2^(2*n - 2)*binomial(n - 3/2, -1/2)/((n + 1)*n):
seq(a(n), n = 1..22); # Peter Luschny, May 12 2021
MATHEMATICA
Array[3 (2 #^3 + # - 1)*2^(2 # - 2)*Binomial[# - 3/2, -1/2]/(# (# + 1)) &, 22] (* Michael De Vlieger, Jan 17 2024 *)
PROG
(Sage)
def a(n):
return binomial(2*n, n)*(2*n+1)/2+n*binomial(2*n-2, n)+(n-1)*binomial(2*n-2, n+1)
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, May 12 2021
STATUS
approved