

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
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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



FORMULA

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.
Dfinite with recurrence (n^35*n^26*n)*a(n)+(6*n^3+27*n^2+35*n+12)*a(n+1)+(n^34*n^23*n)*a(n+2) = 0.  Robert Israel, Nov 18 2016


EXAMPLE

A convex 3gon 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 4gon 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^26*z+1)^(1/2): g := z/s^3(1/s(z+1s)/(4*z))/2: ser := series(g, z, 30):


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.


KEYWORD

nonn,frac


AUTHOR



STATUS

approved



