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

0,3

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

Table of n, a(n) for n=0..23.

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.

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

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

Prog

(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)])

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.

Sequence in context: A319959 A270007 A354558 * A166789 A081901 A283686

Adjacent sequences: A344318 A344319 A344320 * A344322 A344323 A344324

Keyword

nonn

Author

F. Chapoton, May 15 2021

Extensions

Better name from Peter Luschny, May 16 2021

Status

approved