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

0,3

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.

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.

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

Example

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

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 *)

Crossrefs

Cf. A128751.

Sequence in context: A273461 A217190 A020698 * A074611 A362381 A292078

Adjacent sequences: A128749 A128750 A128751 * A128753 A128754 A128755

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved