A344191 -id:A344191 - cp's OEIS frontend

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

F. Chapoton, May 12 2021

nonn

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.

			For B_2, among the 18 intervals in the hexagon-shaped lattice, only one is not linear.

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.

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

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

Array[3 (2 #^3 + # - 1)*2^(2 # - 2)*Binomial[# - 3/2, -1/2]/(# (# + 1)) &, 22] (* Michael De Vlieger, Jan 17 2024 *)

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)

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

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.

Original entry on oeis.org

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

A344216 a(n) = n!((n+1)/2 + 2Sum_{k=2..n-1}(n-k)/(k+1)).

Original entry on oeis.org

1, 3, 16, 104, 768, 6336, 57888, 581472, 6379200, 75977280, 977045760, 13499930880, 199537067520, 3142504512000, 52546707763200, 929908914278400, 17366044153651200, 341336836618444800, 7044417438363648000

Offset: 1

F. Chapoton, May 13 2021

nonn

Conjecture: a(n) is the number of linear intervals in the weak order on the symmetric group S_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to a(7) = 57888.

			For S_3, among the 17 intervals in the hexagon-shaped lattice, only the full lattice is not linear.

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. 145.

Cf. A344136, A344191, A344228 for similar sequences.
Cf. A007767 for all intervals in the weak order on S_n.
Cf. A001008, A002805.

a := n -> (1/2)*n!*(4*(n + 1)*harmonic(n) - 9*n + 3):
# Or:
egf := (3 - 8*x - 4*ln(1 - x))/(2*(x - 1)^2):
ser := series(egf, x, 24): a := n -> n!*coeff(ser, x, n):
seq(a(n), n=1..19); # Peter Luschny, May 13 2021

Join[{1}, RecurrenceTable[{(n - 3) a[n] == (2 n^2 - 5 n - 1) a[n - 1] - (n^3 - 3 n^2 + 2 n) a[n - 2], a[2] == 3, a[3] == 16}, a, {n, 2, 19}]] (* Peter Luschny, May 13 2021 *)

a(n) = n!*((n+1)/2+2*sum(k=2, n-1, (n-k)/(k+1))); \\ Michel Marcus, May 13 2021

def a(n):
    return factorial(n)*((n+1)/2+2*sum((n-k)/(k+1) for k in range(2, n)))

From Peter Luschny, May 13 2021: (Start)
a(n) = (1/2) * n! * (4 * (n + 1) * H(n) - 9*n + 3), where H(n) are the harmonic numbers H(n) = A001008(n)/A002805(n).
a(n) = n! * [x^n] (3 - 8*x - 4*log(1 - x))/(2*(x - 1)^2).
a(n) = ((2*n^2 - 5*n - 1)*a(n-1) - (n^3 - 3*n^2 + 2*n)*a(n-2))/(n - 3) for n >= 4. (End)

A034275 a(n) = f(n,n-2) where f is given in A034261.

Original entry on oeis.org

1, 3, 14, 65, 294, 1302, 5676, 24453, 104390, 442442, 1864356, 7818538, 32657884, 135950700, 564306840, 2336457645, 9652643910, 39800950530, 163830074100, 673327275390, 2763494696820, 11327881630260, 46381659765480, 189711966348450, 775232392541724, 3165127107345252

Offset: 1

Clark Kimberling

Divisible by the Catalan numbers, by the explicit formula. - F. Chapoton, Jun 24 2021

Cf. A000108, A002061, A034261, A344191, A344228.

a[n_] := Binomial[2*n-2,n-1] * (n^2-n+1) / n; Array[a, 25] (* Amiram Eldar, Sep 04 2025 *)

a(n) = binomial(2*n-2,n-1)/n * (n^2-n+1); \\ Michel Marcus, Jun 24 2021

[binomial(2*n-2,n-1)//n * (n**2-n+1) for n in range(1,8)]

a(n) = binomial(2*n-2,n-1)/n * (n^2-n+1).
a(n) = binomial(2*n-2,n-1) + (n-1)*binomial(2*n-2,n).
D-finite with recurrence n*a(n) + 2*(-6*n+7)*a(n-1) + 4*(11*n-24)*a(n-2) + 24*(-2*n+7)*a(n-3) = 0. - R. J. Mathar, Feb 10 2025
a(n) ~ 2^(2*n-2) * sqrt(n/Pi). - Amiram Eldar, Sep 04 2025

Corrected and extended by N. J. A. Sloane, Apr 21 2000

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

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).

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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.

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A344216 a(n) = n!*((n+1)/2 + 2*Sum_{k=2..n-1}(n-k)/(k+1)).

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A034275 a(n) = f(n,n-2) where f is given in A034261.

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

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

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

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.

A344216 a(n) = n!((n+1)/2 + 2Sum_{k=2..n-1}(n-k)/(k+1)).

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