A128750 Number of skew Dyck paths of semilength n having no ascents of length 1.

1, 0, 2, 4, 14, 44, 150, 520, 1850, 6696, 24602, 91500, 343846, 1303572, 4979822, 19150352, 74075890, 288022160, 1125076210, 4413061972, 17375007294, 68641377980, 272014578822, 1081009104664, 4307221752874, 17203123381304

Offset: 0

Author

Emeric Deutsch, Mar 31 2007

Keywords

• nonn

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.

Hankel transform is 2^ceiling(n(n+1)/3). Binomial transform is A059278. - Paul Barry, Feb 11 2009

Examples

			a(3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Cf. A128749.

Programs

Maple

G:=(1-z^2-sqrt((1-z^2)*(1-4*z-z^2)))/2/z/(1+z): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..30);

Mathematica

CoefficientList[Series[(1-x^2-Sqrt[(1-x^2)*(1-4*x-x^2)])/2/x/(1+x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

a(n) = A128749(n,0).

G.f.: G = G(z) satisfies z(1 + z)G^2 - (1 - z^2)G + 1 - z = 0.

G.f.: 1/(1+x-x/(1-x-x/(1+x-x/(1-x-x/(1+x-x/(1-... (continued fraction). - Paul Barry, Feb 11 2009

From Paul Barry, Feb 11 2009: (Start)

G.f.: (1/(1+x))c(x/(1-x^2)) where c(x) is the g.f. of A000108;

G.f.: 1/(1-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-.... (continued fraction);

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(floor((n+k)/2),k)*A000108(k).

(End)

Conjecture: (n+1)*a(n) +(-4*n+3)*a(n-1) +(-2*n-1)*a(n-2) +(4*n-11)*a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2012

a(n) ~ sqrt(5+3*sqrt(5)) * (2+sqrt(5))^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014