A257887 Number of proper diagonals of the n-dimensional associahedron (i.e., diagonals that are not included in lower dimension faces).

1, 5, 34, 273, 2436, 23391, 237090, 2505228, 27360612, 306956091, 3521389998, 41164654020, 489017000736, 5890746106977, 71829149873286, 885296835708778, 11015753148497480, 138241674405266782, 1748203287998505712, 22261537862360050040, 285268915333307553016

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

Sequence in context: A058248 A116435 A292877 * A090367 A189488 A111557

Adjacent sequences: A257884 A257885 A257886 * A257888 A257889 A257890

Keyword

nonn

Author

Alon Regev, May 11 2015

Extensions

More terms from F. Chapoton, Feb 17 2017

Status

approved