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 A277652 Numerators of factorial moments of order 2 for the number of parts in dissections of rooted and convex polygons. 1
 0, 0, 4, 40, 312, 2212, 14920, 97632, 626080, 3957448, 24747948, 153483720, 945638232, 5795135820, 35357242128, 214919392128, 1302250826880, 7869116134672, 47437683195220, 285373276253352, 1713562776624952, 10272384482513140, 61489533128765784, 367581030765071200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n)/A001003(n) is the factorial moment of order two for the number of parts in a (uniform) random (rooted) dissection of a convex (n+2)-gon. LINKS Robert Israel, Table of n, a(n) for n = 0..1300 Ricardo Gómez Aíza, RNA structures and dissections of polygons: an invitation to analytic combinatorics, Misc. Mat. 60 (2015) 105-130 (In Spanish) FORMULA a(n) = A002695(n) - A035029(n-1), n >= 1. G.f.: (z/sqrt(z^2 - 6*z + 1)^3) - (1/sqrt(z^2 - 6*z + 1) - (z + 1 - sqrt(z^2 - 6*z + 1))/(4*z))/2. D-finite with recurrence (-n^3-5*n^2-6*n)*a(n)+(6*n^3+27*n^2+35*n+12)*a(n+1)+(-n^3-4*n^2-3*n)*a(n+2) = 0. - Robert Israel, Nov 18 2016 EXAMPLE A convex 3-gon is a triangle. There is only one dissection of a rooted triangle, with one single part. The factorial moment of order two is therefore 0 and hence a(1) = 0. A convex 4-gon is a quadrilateral. There are three dissections of a rooted quadrilateral, two with two parts and one with one part. Then the expectation of the number of parts is 5/3, and the expectation of the number of parts squared is 9/3, hence the factorial moment of order two is 9/3 - 5/3 = 4/3. The second Schröder number is A001003(2) = 3, therefore a(2) = 4. MAPLE s := (z^2-6*z+1)^(1/2): g := z/s^3-(1/s-(z+1-s)/(4*z))/2: ser := series(g, z, 30): seq(coeff(ser, z, n), n=0..23); # Peter Luschny, Nov 17 2016 MATHEMATICA CoefficientList[Series[z/Sqrt[(z^2 - 6*z + 1)^3] - (1/Sqrt[z^2 - 6*z + 1] - (z + 1 - Sqrt[z^2 - 6*z + 1])/(4*z))/2, {z, 0, 20}], z] CROSSREFS Denominators are the Schröder numbers A001003. Cf. A002695, A035029. Sequence in context: A366667 A212699 A061318 * A190541 A298198 A043031 Adjacent sequences: A277649 A277650 A277651 * A277653 A277654 A277655 KEYWORD nonn,frac AUTHOR Ricardo Gómez Aíza, Oct 26 2016 STATUS approved

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Last modified April 18 12:49 EDT 2024. Contains 371780 sequences. (Running on oeis4.)