A129160 Sum of the semi-abscissae of the first returns to the axis over all skew Dyck paths of semilength n.

1, 4, 18, 82, 378, 1760, 8262, 39044, 185526, 885596, 4243590, 20400954, 98353278, 475322352, 2302064010, 11170370850, 54293503770, 264290420540, 1288257980310, 6287181414470, 30717958762350, 150234512678480, 735446569221810, 3603330368706640, 17668505697688098, 86698739895529300

Offset

1,2

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 the path is defined to be the number of its steps.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from Vincenzo Librandi)

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Formula

a(n) = Sum_{k=1,..,n} k*A129159(n,k).

a(n) = 2*A128752(n) for n>=2.

G.f.: x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^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, Oct 20 2012

a(n) ~ 6*5^(n-3/2)/sqrt(Pi*n) . - Vaclav Kotesovec, Oct 20 2012

Example

a(2)=4 because UDUD, UUDD and UUDL yield 1+2+1=4.

Maple

G:=z-1+(1-3*z+2*z^2)/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);

Mathematica

CoefficientList[Series[(1/x) (x - 1 + (1 - 3*x + 2*x^2)/Sqrt[1 - 6*x + 5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

Prog

(PARI) x='x+O('x^25); Vec(x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2)) \\ G. C. Greubel, Feb 09 2017

Crossrefs

Cf. A129159, A128752.

Sequence in context: A100192 A052913 A279285 * A187077 A218986 A143646

Adjacent sequences: A129157 A129158 A129159 * A129161 A129162 A129163

Keyword

nonn

Author

Emeric Deutsch, Apr 03 2007

Extensions

Mathematica code corrected by Vincenzo Librandi, May 24 2013

Status

approved