A114627 Number of hill-free Dyck paths of semilength n+3 and having no peaks at level 2 (a Dyck path is said to be hill-free if it has no peaks at level 1).

1, 2, 6, 19, 61, 202, 683, 2348, 8184, 28855, 102731, 368813, 1333684, 4853436, 17761181, 65320691, 241300829, 894958140, 3331323651, 12441078958, 46601721324, 175040968111, 659136721385, 2487852579751, 9410480922018

Offset

0,2

Comments

Column 0 of A114626.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: (C-1)/(z*(1+z+z^2-z*(1+z)*C), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.

a(n) ~ 4^(n+5) / (121*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence +(n+3)*a(n) +(-n+3)*a(n-1) +2*(-5*n-6)*a(n-2) +(-7*n-9)*a(n-3) +2*(-2*n-3)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

Example

a(1)=2 because we have UUUDUDDD and UUUUDDDD, where U=(1,1), D=(1,-1).

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=(C-1)/z/(1+z+z^2-z*(1+z)*C): Gser:=series(G, z=0, 32): 1, seq(coeff(Gser, z^n), n=1..28);

Mathematica

CoefficientList[Series[((1-Sqrt[1-4*x])/2/x-1)/x/(1+x+x^2-x*(1+x)*(1-Sqrt[1-4*x])/2/x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Prog

(PARI) x='x+O('x^50); Vec((1-2*x-sqrt(1-4*x))/(x^2*(2*x^2+x+1+(1+x)*sqrt(1-4*x)))) \\ G. C. Greubel, Mar 18 2017

Crossrefs

Cf. A114626.

Sequence in context: A228180 A035929 A071646 * A289591 A148464 A148465

Adjacent sequences: A114624 A114625 A114626 * A114628 A114629 A114630

Keyword

nonn

Author

Emeric Deutsch, Dec 18 2005

Status

approved