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.

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

Author

F. Chapoton, May 15 2021

Keywords

• nonn

Comments

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.

Links

• 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.

Crossrefs

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.

Programs

Maple

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

Sage

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)])

Formula

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

Extensions

Better name from Peter Luschny, May 16 2021