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A128752 Number of ascents of length at least 2 in all skew Dyck paths of semilength n. 3
0, 0, 2, 9, 41, 189, 880, 4131, 19522, 92763, 442798, 2121795, 10200477, 49176639, 237661176, 1151032005, 5585185425, 27146751885, 132145210270, 644128990155, 3143590707235, 15358979381175, 75117256339240, 367723284610905 (list; graph; refs; listen; history; text; internal format)
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 A292078 A270766

Adjacent sequences:  A128749 A128750 A128751 * A128753 A128754 A128755

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified April 22 22:14 EDT 2019. Contains 322378 sequences. (Running on oeis4.)