A344216 -id:A344216 - cp's OEIS frontend

A344321 a(n) = 2^(2n - 5)binomial(n-5/2, -1/2)(36n^4 - 78n^3 + 54n^2 - 48n + 24)/((n + 1)n*(n - 1)) for n >= 2 and otherwise 1.

1, 1, 8, 49, 246, 1157, 5248, 23256, 101398, 436865, 1865136, 7906054, 33319388, 139754994, 583859968, 2430991670, 10092510630, 41794856985, 172699266480, 712220712390, 2932169392020, 12052941519030, 49475929052160, 202838118604680

Offset: 0

F. Chapoton, May 15 2021

nonn

Conjecture: These are the number of linear intervals in the Cambrian lattices 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 a(8) = 101398.

The term a(3) = 49 is the same as the 49 appearing in A344136.

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.
Peter Luschny, Remark regarding A344228 and A344321.

Cf. A344136 for the type A, A344228 for the type B.
Cf. also A344191, A344216 for similar sequences.
Cf. A344400 and A344401 for an alternative approach.
Cf. A007531.

a := n -> if n < 2 then 1 else 2^(2*n - 5)*binomial(n - 5/2, -1/2)*(36*n^4 - 78*n^3 + 54*n^2 - 48*n + 24)/((n + 1)*n*(n - 1)) fi;
seq(a(n), n = 0..23); # Peter Luschny, May 16 2021

def a(n):
    if n < 2: return 1
    if n == 2: return 8
    return (3*n-2)*(1/n+1/2)*binomial(2*n-2,n-1)+6*(n-2)*binomial(2*n-4,n-2)+(n-1)*(3*n-8)/2/(2*n-3)*binomial(2*n-2,n-1)+sum(2*binomial(k,n-1)*(n+1+k) for k in range(n-1,2*n-5))
print([a(n) for n in range(24)])

a(n) = (3*n-2)*(1/n+1/2)*binomial(2*n-2,n-1) + 6*(n-2)*binomial(2*n-4,n-2) + (n-1)*(3*n-8)/(2*(2*n-3))*binomial(2*n-2,n-1) + 2 Sum_{k=1..2n-6} binomial(k,n-1)*(n+1+k) for n >= 3.

a(n) = A344401(n) / A007531(n+3) for n >= 2. - Peter Luschny, May 17 2021

Better name from Peter Luschny, May 16 2021

A344717 a(n) = (3n - 9/2 - 1/n + 6/(n+1))*binomial(2n-2,n-1).

Original entry on oeis.org

6, 34, 169, 791, 3576, 15807, 68783, 295867, 1261468, 5341128, 22487906, 94244294, 393439840, 1637091585, 6792664635, 28115240595, 116120791380, 478689505140, 1969993524510, 8095052323410, 33218808108720, 136148925337230, 557389537873974, 2279607910207326

Offset: 2

F. Chapoton, May 27 2021

nonn

Conjecture: These are the number of linear intervals in the tilting posets 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 295867 for n = 9.

Michael De Vlieger, Table of n, a(n) for n = 2..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. 152.

For the tilting posets of type A, see A344136.
For the Cambrian lattices of types A, B and D, see A344136, A344228, A344321.
For similar sequences, see A344191, A344216.

Array[(3 # - 9/2 - 1/# + 6/(# + 1))*Binomial[2 # - 2, # - 1] &, 24, 2] (* Michael De Vlieger, Jan 17 2024, after Sage *)

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

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

Original entry on oeis.org

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

Offset: 3

F. Chapoton, May 27 2021

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.

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.

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.

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

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

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)

cp's OEIS Frontend

A344321 a(n) = 2^(2n - 5)binomial(n-5/2, -1/2)(36n^4 - 78n^3 + 54n^2 - 48n + 24)/((n + 1)n*(n - 1)) for n >= 2 and otherwise 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Sage

Formula

Extensions

A344717 a(n) = (3n - 9/2 - 1/n + 6/(n+1))*binomial(2n-2,n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

cp's OEIS Frontend

A344321 a(n) = 2^(2*n - 5)*binomial(n-5/2, -1/2)*(36*n^4 - 78*n^3 + 54*n^2 - 48*n + 24)/((n + 1)*n*(n - 1)) for n >= 2 and otherwise 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Sage

Formula

Extensions

A344717 a(n) = (3n - 9/2 - 1/n + 6/(n+1))*binomial(2n-2,n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A344321 a(n) = 2^(2n - 5)binomial(n-5/2, -1/2)(36n^4 - 78n^3 + 54n^2 - 48n + 24)/((n + 1)n*(n - 1)) for n >= 2 and otherwise 1.

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