A344228 a(n) = binomial(2*n,n)*(2*n+1)/2+n*binomial(2*n-2,n)+(n-1)*binomial(2*n-2,n+1).

3, 17, 84, 393, 1778, 7866, 34254, 147433, 628914, 2663934, 11219728, 47033322, 196393044, 817338580, 3391858530, 14040986985, 57998364690, 239112756630, 984126777480, 4044255577230, 16597080112860, 68027923573740

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

Cf. A344136 for the type A, A344191 for a similar sequence.

Sequence in context: A083217 A225307 A318769 * A216681 A350456 A037787

Adjacent sequences: A344225 A344226 A344227 * A344229 A344230 A344231

Keyword

nonn

Author

F. Chapoton, May 12 2021

Status

approved