

A268208


Number of paths from (0,0) to (n,n) using only steps North, Northeast and East (i.e., steps E(1,0), D(1,1), and N(0,1)) that do not cross y=x "vertically".


0



1, 3, 12, 52, 236, 1108, 5340, 26276, 131484, 667108, 3424108, 17748564, 92776716, 488527284, 2588907708, 13797337668, 73901315644, 397609958596, 2147904635340, 11645489540468, 63349140877356, 345651184335892, 1891209255293852
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

In Pan and Remmel's link, "vertical" crossing is defined via paired pattern P_1 and P_2.


LINKS

Table of n, a(n) for n=0..22.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.


FORMULA

G.f.: (x1)*(1+3*x+sqrt(16*x+x^2))/(x^2*(3x+sqrt(16*x+x^2))).
Conjecture: (n+2)*a(n) +(7*n2)*a(n1) +(7*n16)*a(n2) +(n+4)*a(n3)=0.  R. J. Mathar, Jun 07 2016


EXAMPLE

For example, ENDNE crosses y=x vertically. DDNE does not cross y=x. NEDEN crosses y=x horizontally.
For n=2, there are 13 paths from (0,0) to (2,2) and only one of them crosses y=x vertically, namely ENNE. Therefore, a(2) = 12.


PROG

(PARI) x = 'x + O('x^30); Vec((x1)*(1+3*x+sqrt(16*x+x^2))/(x^2*(3x+sqrt(16*x+x^2)))) \\ Michel Marcus, Feb 02 2016


CROSSREFS

Cf. A001850, A001003, A006318.
Sequence in context: A151192 A151193 A151194 * A007856 A151195 A151196
Adjacent sequences: A268205 A268206 A268207 * A268209 A268210 A268211


KEYWORD

nonn


AUTHOR

Ran Pan, Jan 28 2016


STATUS

approved



