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A257887
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Number of proper diagonals of the n-dimensional associahedron (i.e., diagonals that are not included in lower dimension faces).
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5
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1, 5, 34, 273, 2436, 23391, 237090, 2505228, 27360612, 306956091, 3521389998, 41164654020, 489017000736, 5890746106977, 71829149873286, 885296835708778, 11015753148497480, 138241674405266782, 1748203287998505712, 22261537862360050040, 285268915333307553016
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OFFSET
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1,2
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COMMENTS
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a(n) is also the number of pairs of triangulations of an (n+3)-gon that have no diagonals in common.
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REFERENCES
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D. Huguet and D. Tamari, La structure polyedrale des complexes de parenthesages, J. Combinatorics, Information & System Sciences 3 (1978) pages 69-81
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LINKS
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Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana, Stephan Wagner, Counting Planar Tanglegrams, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.
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MATHEMATICA
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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 *)
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PROG
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(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()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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