OFFSET
0,3
COMMENTS
Conjecture: These are the number of linear intervals in Pallo's comb posets. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term 36686 for n = 9.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1664
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
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. 150.
J. M. Pallo, Right-arm rotation distance between binary trees, Inform. Process. Lett., 87(4):173-177, 2003.
FORMULA
a(n) = Catalan(n) + (1/(n + 2))*Sum_{k=2..n}((2^(n - k)*(n - k + 4)/(k - 2)!)* Product_{i=2..k-1}(n + i)).
From Peter Luschny, May 11 2021: (Start)
a(n) = [x^n] ((2*x + sqrt(1 - 4*x) - 1)*(3*x - 1))/(2*sqrt(1 - 4*x)*x^2).
a(n) = n! * [x^n] exp(2*x)*(BesselI(0, 2*x) - BesselI(1, 2*x) + BesselI(2, 2*x)).
a(n) = a(n-1)*(2*(2*n - 1)*(n^2 + 2))/((n + 2)*(n^2 - 2*n + 3)) for n >= 1.
a(n) ~ (2^(2*n - 3)*(8*n - 25)) / (sqrt(Pi)*n^(3/2)). (End)
a(n) = A121686(n) / 2. - Hugo Pfoertner, May 11 2021
EXAMPLE
All 3 intervals in the poset of cardinality 2 are linear. All 11 intervals in the poset of cardinality 5 are linear.
MAPLE
a := n -> `if`(n = 0, 1, a(n-1)*(2*(2*n-1)*(n^2+2))/((n+2)*(n^2-2*n+3))):
seq(a(n), n = 0..19); # Peter Luschny, May 11 2021
MATHEMATICA
a[n_] := CatalanNumber[n] (n^2 + 2) / (n + 2);
Table[a[n], { n, 0, 23}] (* Peter Luschny, May 11 2021 *)
PROG
(Sage)
def a(n):
return catalan_number(n)+sum(2**(n-k)/factorial(k-2)*(n-k+4)/(n+2)*prod(n+i for i in range(2, k)) for k in range(2, n+1))
(Sage)
def a(n): return catalan_number(n) + binomial(2*n, n-2)
print([a(n) for n in range(24)]) # Peter Luschny, May 11 2021
(PARI) a(n) = (binomial(2*n, n)/(n+1))*((n^2 + 2)/(n + 2)); \\ Michel Marcus, May 11 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, May 11 2021
STATUS
approved