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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.: (x-1)*(-1+3*x+sqrt(1-6*x+x^2))/(x^2*(3-x+sqrt(1-6*x+x^2))).

Conjecture: (n+2)*a(n) +(-7*n-2)*a(n-1) +(7*n-16)*a(n-2) +(-n+4)*a(n-3)=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((x-1)*(-1+3*x+sqrt(1-6*x+x^2))/(x^2*(3-x+sqrt(1-6*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

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.