A277652 Numerators of factorial moments of order 2 for the number of parts in dissections of rooted and convex polygons.

0, 0, 4, 40, 312, 2212, 14920, 97632, 626080, 3957448, 24747948, 153483720, 945638232, 5795135820, 35357242128, 214919392128, 1302250826880, 7869116134672, 47437683195220, 285373276253352, 1713562776624952, 10272384482513140, 61489533128765784, 367581030765071200

Offset: 0

Author

Ricardo Gómez Aíza, Oct 26 2016

Keywords

• nonn
• frac

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.

Examples

			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.
		

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)

Crossrefs

Denominators are the Schröder numbers A001003.

Cf. A002695, A035029.

Programs

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]

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