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
Keywords
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.
Programs
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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 *) -
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
Extensions
More terms from F. Chapoton, Feb 17 2017
Comments