A257887 - OEIS

%I #25 Apr 14 2023 14:23:03

%S 1,5,34,273,2436,23391,237090,2505228,27360612,306956091,3521389998,

%T 41164654020,489017000736,5890746106977,71829149873286,

%U 885296835708778,11015753148497480,138241674405266782,1748203287998505712,22261537862360050040,285268915333307553016

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

%C a(n) is also the number of pairs of triangulations of an (n+3)-gon that have no diagonals in common.

%D D. Huguet and D. Tamari, La structure polyedrale des complexes de parenthesages, J. Combinatorics, Information & System Sciences 3 (1978) pages 69-81

%H Andrew Howroyd, <a href="/A257887/b257887.txt">Table of n, a(n) for n = 1..200</a>

%H Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, <a href="https://arxiv.org/abs/2304.05318">Sampling planar tanglegrams and pairs of disjoint triangulations</a>, arXiv:2304.05318 [math.CO], 2023.

%H Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana, Stephan Wagner, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.32">Counting Planar Tanglegrams</a>, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.

%t 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_ *)

%o (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

%o (Sage)

%o x = PowerSeriesRing(QQ, 'x').gen()

%o N = 30

%o A = sum(catalan_number(n) ** 2 * x ** (n + 1) for n in range(N)).O(N + 1)

%o B = A.reverse()

%o list((-B+x-x**2)/2) # _F. Chapoton_, Feb 17 2017

%K nonn

%O 1,2

%A _Alon Regev_, May 11 2015

%E More terms from _F. Chapoton_, Feb 17 2017