

A128723


Number of skew Dyck paths of semilength n having no peaks at level 1.


2



1, 0, 2, 6, 22, 84, 334, 1368, 5734, 24480, 106086, 465462, 2063658, 9231084, 41610162, 188820726, 861891478, 3954732384, 18230522422, 84390187986, 392120098258, 1828220666844, 8550445900442, 40103716079436
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OFFSET

0,3


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 a path is defined to be the number of steps in it.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

a(n) = A128722(n,0).
a(n) = 2*A117641(n) for n>=1.
G.f.: (33*zsqrt(16*z+5*z^2))/(1+3*z+sqrt(16*z+5*z^2)).
a(n) ~ 5^(n+3/2)/(16*sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Mar 20 2014
Conjecture: +3*(n+1)*a(n) +(17*n+10)*a(n1) +9*(n3)*a(n2) +5*(n2)*a(n3)=0.  R. J. Mathar, Jun 17 2016


EXAMPLE

a(3)=6 because we have UUDUDD, UUUDDD, UUUDLD, UUDUDL, UUUDDL and UUUDLL.


MAPLE

G:=(33*zsqrt(16*z+5*z^2))/(1+3*z+sqrt(16*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);


MATHEMATICA

CoefficientList[Series[(33*xSqrt[16*x+5*x^2])/(1+3*x+Sqrt[16*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG

(PARI) x='x+O('x^50); Vec((33*zsqrt(16*z+5*z^2)) /(1+3*z +sqrt(16*z+5*z^2))) \\ G. C. Greubel, Mar 19 2017


CROSSREFS

Cf. A117641, A128722.
Sequence in context: A164870 A121686 A245904 * A150244 A151288 A150245
Adjacent sequences: A128720 A128721 A128722 * A128724 A128725 A128726


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 31 2007


STATUS

approved



