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

0,4

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

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.

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

Example

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

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 *)

Crossrefs

Cf. A001850, A110121.

Sequence in context: A053464 A111998 A026935 * A233657 A346842 A081017

Adjacent sequences: A110124 A110125 A110126 * A110128 A110129 A110130

Keyword

nonn

Author

Emeric Deutsch, Jul 13 2005

Status

approved