OFFSET
0,2
COMMENTS
In Pan and Remmel's link, "vertical" crossing is defined via paired pattern P_1 and P_2.
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Ran Pan, Jan 28 2016
STATUS
approved