login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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.
Sequence in context: A366667 A212699 A061318 * A190541 A298198 A043031
KEYWORD
nonn,frac
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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