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".

1, 3, 12, 52, 236, 1108, 5340, 26276, 131484, 667108, 3424108, 17748564, 92776716, 488527284, 2588907708, 13797337668, 73901315644, 397609958596, 2147904635340, 11645489540468, 63349140877356, 345651184335892, 1891209255293852

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.

Luis Verde-Star A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Formula

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

D-finite with recurrence (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

a(n) = Sum_{m=0..n} C(2*m+2,m)*C(m+n,n-m)/(m+1). - Vladimir Kruchinin Jan 20 2021

a(n) ~ 2^(5/4) * (1 + sqrt(2))^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 20 2021

a(n) = hypergeom([3/2, -n, n + 1], [1/2, 3], -1). - Peter Luschny, Jan 20 2021

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.

Maple

a := n -> hypergeom([3/2, -n, n + 1], [1/2, 3], -1):
seq(simplify(a(n)), n=0..22); # Peter Luschny, Jan 20 2021

Prog

(PARI) my(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
(Maxima)
a(n):=sum(((binomial(2*m+2, m))*(binomial(m+n, n-m)))/(m+1), m, 0, n); /* Vladimir Kruchinin, Jan 20 2021 */

Crossrefs

Cf. A001850, A001003, A006318.

Sequence in context: A151192 A151193 A151194 * A007856 A151195 A151196

Adjacent sequences: A268205 A268206 A268207 * A268209 A268210 A268211

Keyword

nonn,easy

Author

Ran Pan, Jan 28 2016

Status

approved