

A129160


Sum of the semiabscissae of the first returns to the axis over all skew Dyck paths of semilength n.


2



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
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OFFSET

1,2


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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) 21912203


FORMULA

a(n) = Sum_{k=1,..,n} k*A129159(n,k).
a(n) = 2*A128752(n) for n>=2.
G.f.: x1+(13*x+2*x^2)/sqrt(16*x+5*x^2).
Recurrence: n*(3*n1)*a(n) = 18*(n1)*n*a(n1)  5*(n3)*(3*n+2)*a(n2) .  Vaclav Kotesovec, Oct 20 2012
a(n) ~ 6*5^(n3/2)/sqrt(Pi*n) .  Vaclav Kotesovec, Oct 20 2012


EXAMPLE

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


MAPLE

G:=z1+(13*z+2*z^2)/sqrt(16*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(x1+(13*x+2*x^2)/sqrt(16*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



