A344400 -id:A344400 - 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

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

A344399 a(n) = 4^nbinomial(n - 1/2, -1/2)(n^2 + 1).

Original entry on oeis.org

1, 4, 30, 200, 1190, 6552, 34188, 171600, 836550, 3986840, 18660356, 86062704, 392102620, 1768102000, 7902970200, 35056559520, 154477660230, 676745803800, 2949418972500, 12794985495600, 55276458056820, 237909980502480, 1020487997404200, 4363718285628000

Offset: 0

Peter Luschny, May 17 2021

Cf. A344400, A344401.

aList := proc(len) local gf, ser;
gf := (20*x^2 - 6*x + 1) / (1 - 4*x)^(5/2): ser := series(gf, x, len+2):
seq(coeff(ser, x, n), n = 0..len) end: aList(23);

Table[4^n Binomial[n-1/2,-1/2](n^2+1),{n,0,30}] (* Harvey P. Dale, Jun 20 2021 *)

a(n) = [x^n] (20*x^2 - 6*x + 1) / (1 - 4*x)^(5/2).
a(n) = a(n-1)*(-2 + 4*n - 2*n^2 + 4*n^3) / (2*n - 2*n^2 + n^3) for n > 0.
a(n) ~ 4^n * n^(3/2) / sqrt(Pi). - Amiram Eldar, Sep 07 2025

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A344399 a(n) = 4^nbinomial(n - 1/2, -1/2)(n^2 + 1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A344399 a(n) = 4^n*binomial(n - 1/2, -1/2)*(n^2 + 1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

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.

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

A344399 a(n) = 4^nbinomial(n - 1/2, -1/2)(n^2 + 1).