A110127 Number of EE's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1) in all Delannoy paths of length n.

0, 0, 1, 10, 75, 508, 3277, 20566, 126871, 773688, 4679769, 28136546, 168395235, 1004239156, 5971820709, 35429993390, 209800355631, 1240361694064, 7323260678065, 43187703202234, 254439363998587, 1497730375793004

Offset: 0

Author

Emeric Deutsch, Jul 13 2005

Keywords

• nonn

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

{A110127}[n+2] = conv({0, {A002002})[n]. - Tilman Neumann, Feb 05 2009

Examples

			a(2) = 1 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y = x.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Crossrefs

Cf. A001850, A110121.

Programs

Maple

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z^2*R^2/(1-6*z+z^2): Gser:=series(G,z=0,27): 0,seq(coeff(Gser,z^n),n=1..24);

Mathematica

CoefficientList[Series[x^2*((1-x-Sqrt[1-6*x+x^2])/2/x)^2/(1-6*x+x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} k*A110121(n,k).

G.f.: z^2*R^2/(1-6z+z^2), where R = 1+zR+zR^2 = [1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).

Recurrence: n*(2*n-5)*a(n) = 6*(4*n^2 - 13*n + 8)*a(n-1) - 4*(19*n^2 - 76*n + 75)*a(n-2) + 6*(4*n^2 - 19*n + 20)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 1/8*sqrt(2)*(3+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 24 2012