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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. 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,changed AUTHOR Ran Pan, Jan 28 2016 STATUS approved

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Last modified August 4 11:18 EDT 2024. Contains 374920 sequences. (Running on oeis4.)