OFFSET
1,2
COMMENTS
a(n) is also the number of pairs of triangulations of an (n+3)-gon that have no diagonals in common.
REFERENCES
D. Huguet and D. Tamari, La structure polyedrale des complexes de parenthesages, J. Combinatorics, Information & System Sciences 3 (1978) pages 69-81
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana, Stephan Wagner, Counting Planar Tanglegrams, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.
MATHEMATICA
nmax = 23; A = Sum[CatalanNumber[n]^2*x^(n+1), {n, 0, nmax}]+O[x]^(nmax+1); B = InverseSeries[A, x] // Normal; Drop[CoefficientList[(-B+x-x^2)/2, x], 3] (* Jean-François Alcover, Feb 20 2017, after F. Chapoton *)
PROG
(PARI) seq(n)={Vec(x - x^2 - serreverse(sum(k=0, n+1, (binomial(2*k, k)/(k+1))^2*x^(k+1)) + O(x^(n+3))))/2} \\ Andrew Howroyd, Mar 18 2018
(Sage)
x = PowerSeriesRing(QQ, 'x').gen()
N = 30
A = sum(catalan_number(n) ** 2 * x ** (n + 1) for n in range(N)).O(N + 1)
B = A.reverse()
list((-B+x-x**2)/2) # F. Chapoton, Feb 17 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Alon Regev, May 11 2015
EXTENSIONS
More terms from F. Chapoton, Feb 17 2017
STATUS
approved