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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. 3
0, 0, 1, 10, 75, 508, 3277, 20566, 126871, 773688, 4679769, 28136546, 168395235, 1004239156, 5971820709, 35429993390, 209800355631, 1240361694064, 7323260678065, 43187703202234, 254439363998587, 1497730375793004 (list; graph; refs; listen; history; text; internal format)
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 A081017 A238987

Adjacent sequences:  A110124 A110125 A110126 * A110128 A110129 A110130

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 13 2005

STATUS

approved

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Last modified August 15 01:47 EDT 2020. Contains 336485 sequences. (Running on oeis4.)