|
%I #54 Mar 02 2024 03:21:40
%S 0,0,4,40,312,2212,14920,97632,626080,3957448,24747948,153483720,
%T 945638232,5795135820,35357242128,214919392128,1302250826880,
%U 7869116134672,47437683195220,285373276253352,1713562776624952,10272384482513140,61489533128765784,367581030765071200
%N Numerators of factorial moments of order 2 for the number of parts in dissections of rooted and convex polygons.
%C 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.
%H Robert Israel, <a href="/A277652/b277652.txt">Table of n, a(n) for n = 0..1300</a>
%H Ricardo Gómez Aíza, <a href="https://miscelaneamatematica.org/welcome/default/download/tbl_articulos.pdf2.a96de85f16d9d40a.363030372e706466.pdf">RNA structures and dissections of polygons: an invitation to analytic combinatorics</a>, Misc. Mat. 60 (2015) 105-130 (In Spanish)
%F a(n) = A002695(n) - A035029(n-1), n >= 1.
%F 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.
%F 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
%e 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.
%e 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.
%p 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):
%p seq(coeff(ser,z,n), n=0..23); # _Peter Luschny_, Nov 17 2016
%t 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]
%Y Denominators are the Schröder numbers A001003.
%Y Cf. A002695, A035029.
%K nonn,frac
%O 0,3
%A _Ricardo Gómez Aíza_, Oct 26 2016
| 