A344321 -id:A344321 - cp's OEIS frontend

A344400 a(n) = [x^n] 6(8x^3 + 3x + 1) / (1 - 4x)^(7/2).

6, 102, 1008, 7860, 53340, 330372, 1918224, 10615176, 56602260, 293032740, 1481004096, 7337198232, 35743534008, 171641101800, 814046047200, 3819148459920, 17747499595140, 81776562767460, 373968175442400, 1698587342436600, 7667851012141320, 34422129328312440

Offset: 0

Peter Luschny, May 16 2021

nonn

The sequence and its sister sequence A344401 are related to Frédéric Chapoton's sequences A344228 and A344321, as described in the linked remark.

Peter Luschny, Remark regarding A344228 and A344321.

Cf. A344401, A344228, A344321.

aList := proc(len) local gf, ser;
   gf := 6*(8*x^3 + 3*x + 1) / (1 - 4*x)^(7/2):
   ser := series(gf, x, len+2): seq(coeff(ser, x, n), n = 0..len) end:
aList(21);

a(n) = A344228(n+1)*(n+1)*(n+2). - John Keith, May 23 2021

A344401 a(n) = [x^n] 24(-40x^4 + 49x^3 - 15x^2 + 13x + 2) / (1 - 4x)^(9/2).

Original entry on oeis.org

48, 1176, 14760, 138840, 1102080, 7814016, 51104592, 314542800, 1846484640, 10435991280, 57176069808, 305224906896, 1593937712640, 8168132011200, 41177443370400, 204627619798560, 1004073535314720, 4871589672747600, 23398711748319600, 111369179635837200

Offset: 0

Peter Luschny, May 16 2021

nonn

The sequence and its sister sequence A344400 are related to Frédéric Chapoton's sequences A344228 and A344321, as described in the linked remark.

Peter Luschny, Remark regarding A344228 and A344321.

Cf. A344400, A344228, A344321.

aList := proc(len) local gf, ser;
   gf := 24*(-40*x^4 + 49*x^3 - 15*x^2 + 13*x + 2) / (1 - 4*x)^(9/2):
   ser := series(gf, x, len+2): seq(coeff(ser, x, n), n = 0..len) end:
aList(19);

a(n) = if(n==0, 48, 6*(3*n + 4)*(2*n^3 + 9*n^2 + 13*n + 4)*binomial(2*n-1, n)) \\ Andrew Howroyd, May 28 2021

a(n) = 6*(3*n + 4)*(2*n^3 + 9*n^2 + 13*n + 4)*binomial(2*n-1, n) for n>=1. - John Keith, May 28 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

A344400 a(n) = [x^n] 6*(8*x^3 + 3*x + 1) / (1 - 4*x)^(7/2).

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A344401 a(n) = [x^n] 24*(-40*x^4 + 49*x^3 - 15*x^2 + 13*x + 2) / (1 - 4*x)^(9/2).

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Formula

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

A344400 a(n) = [x^n] 6(8x^3 + 3x + 1) / (1 - 4x)^(7/2).

A344401 a(n) = [x^n] 24(-40x^4 + 49x^3 - 15x^2 + 13x + 2) / (1 - 4x)^(9/2).

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