%I #41 Jan 17 2024 10:34:33
%S 1,1,3,11,42,162,627,2431,9438,36686,142766,556206,2169268,8469060,
%T 33096195,129454695,506793270,1985612310,7785510810,30548406570,
%U 119944382220,471241577820,1852521913710,7286586193926,28675561428972,112905199767052,444752335104252
%N a(n) = Catalan(n) * (n^2 + 2) / (n + 2).
%C 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.
%H Michael De Vlieger, <a href="/A344191/b344191.txt">Table of n, a(n) for n = 0..1664</a>
%H Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, <a href="https://ajc.maths.uq.edu.au/pdf/84/ajc_v84_p398.pdf">Dyck Paths with catastrophes modulo the positions of a given pattern</a>, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
%H Clément Chenevière, <a href="https://theses.hal.science/tel-04255439">Enumerative study of intervals in lattices of Tamari type</a>, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 150.
%H J. M. Pallo, <a href="http://dx.doi.org/10.1016/S0020-0190(03)00283-7">Right-arm rotation distance between binary trees</a>, Inform. Process. Lett., 87(4):173-177, 2003.
%F 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)).
%F From _Peter Luschny_, May 11 2021: (Start)
%F a(n) = [x^n] ((2*x + sqrt(1 - 4*x) - 1)*(3*x - 1))/(2*sqrt(1 - 4*x)*x^2).
%F a(n) = n! * [x^n] exp(2*x)*(BesselI(0, 2*x) - BesselI(1, 2*x) + BesselI(2, 2*x)).
%F a(n) = a(n-1)*(2*(2*n - 1)*(n^2 + 2))/((n + 2)*(n^2 - 2*n + 3)) for n >= 1.
%F a(n) = Catalan(n) + binomial(2*n, n-2) = A000108(n) + A002694(n).
%F a(n) ~ (2^(2*n - 3)*(8*n - 25)) / (sqrt(Pi)*n^(3/2)). (End)
%F a(n) = A121686(n) / 2. - _Hugo Pfoertner_, May 11 2021
%e All 3 intervals in the poset of cardinality 2 are linear. All 11 intervals in the poset of cardinality 5 are linear.
%p a := n -> `if`(n = 0, 1, a(n-1)*(2*(2*n-1)*(n^2+2))/((n+2)*(n^2-2*n+3))):
%p seq(a(n), n = 0..19); # _Peter Luschny_, May 11 2021
%t a[n_] := CatalanNumber[n] (n^2 + 2) / (n + 2);
%t Table[a[n], { n, 0, 23}] (* _Peter Luschny_, May 11 2021 *)
%o (Sage)
%o def a(n):
%o 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))
%o (Sage)
%o def a(n): return catalan_number(n) + binomial(2*n, n-2)
%o print([a(n) for n in range(24)]) # _Peter Luschny_, May 11 2021
%o (PARI) a(n) = (binomial(2*n,n)/(n+1))*((n^2 + 2)/(n + 2)); \\ _Michel Marcus_, May 11 2021
%Y Cf. A000108, A002694, A127632, A344136, A121686.
%K nonn
%O 0,3
%A _F. Chapoton_, May 11 2021