A128752 Number of ascents of length at least 2 in all skew Dyck paths of semilength n.

0, 0, 2, 9, 41, 189, 880, 4131, 19522, 92763, 442798, 2121795, 10200477, 49176639, 237661176, 1151032005, 5585185425, 27146751885, 132145210270, 644128990155, 3143590707235, 15358979381175, 75117256339240, 367723284610905

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.

Examples

			a(2)=2 because we have UUDD and UUDL.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Cf. A128751.

Programs

Maple

G:=(1/2)*(1-2*z)*sqrt((1-z)/(1-5*z))-1/2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);

Mathematica

CoefficientList[Series[1/2*(1-2*x)*Sqrt[(1-x)/(1-5*x)]-1/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 20 2012 *)

Formula

a(n) = Sum_{k>=0} k*A128751(n,k).

G.f.: (1/2)(1-2z)sqrt((1-z)/(1-5z)) - 1/2.

Recurrence: n*(3*n-1)*a(n) = 18*(n-1)*n*a(n-1) - 5*(n-3)*(3*n+2)*a(n-2). - Vaclav Kotesovec, Nov 20 2012

a(n) ~ 3*5^(n-3/2)/sqrt(Pi*n). - Vaclav Kotesovec, Nov 20 2012