A344728 a(n) = (9*n/4 - 51/8 - 5/(16*n-24) + 1/n + 6/(n+1))*binomial(2*n-2,n-1).

12, 79, 419, 2036, 9435, 42449, 187187, 813592, 3497988, 14912910, 63151022, 265958200, 1114981465, 4656455685, 19383036675, 80456688240, 333146169840, 1376479675890, 5676426414810, 23369047049400, 96060414949590

Offset: 3

Author

F. Chapoton, May 27 2021

Keywords

• nonn
• easy

Comments

Conjecture: a(n) is the number of linear intervals in the tilting posets of type D_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term 187187 for n = 9.

Links

• Michael De Vlieger, Table of n, a(n) for n = 3..1659
• 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. 152.

Crossrefs

For the tilting posets of types A and B, see A344136, A344717.

For the Cambrian lattices of types A, B and D, see A344136, A344228, A344321.

For similar sequences, see A344191, A344216.

Programs

Mathematica

Array[(9/4 # - 51/8 - 5/8/(2 # - 3) + 1/# + 6/(# + 1))*Binomial[2 # - 2, # - 1] &, 21, 3] (* Michael De Vlieger, Jan 17 2024 *)

PARI

a(n) = (9*n/4-51/8-5/(16*n-24)+1/n+6/(n+1))*binomial(2*n-2,n-1) \\ Felix Fröhlich, May 27 2021

Sage

def a(n):
    return (9/4*n-51/8-5/8/(2*n-3)+1/n+6/(n+1))*binomial(2*n-2,n-1)