A114589 Number of hill-free Dyck paths of semilength n+3 and having no peaks at even levels (a hill in a Dyck path is a peak at level 1).

1, 1, 3, 7, 17, 43, 110, 286, 753, 2003, 5376, 14540, 39589, 108427, 298512, 825664, 2293271, 6393539, 17885835, 50191175, 141247519, 398537101, 1127203038, 3195229662, 9076078057, 25830193513, 73643406563, 210312889095

Offset

0,3

Comments

Column 0 of A114588. The number of hill-free Dyck paths having no peaks at odd level are given by the Riordan numbers (A005043).

From Paul Barry, Jul 05 2009: (Start)

The sequence 1,0,0,1,1,3,7,...

has g.f. ((1+x)*(1+2*x)-sqrt((1+x)*(1-3*x)))/(2*x*(2+2*x+x^2)).

It is the inverse binomial transform of A035929(n+1). (End)

Links

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

Formula

G.f.: (1 -z -2*z^2 -2*z^3 -sqrt(1-3*z^2-2*z))/(2*z^4*(2+2*z+z^2)).

a(n) ~ 3^(n+11/2) / (50*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: 2*(n+4)*a(n) +2*(-n-1)*a(n-1) +3*(-3*n-4)*a(n-2) +(-8*n-11)*a(n-3) +3*(-n-1)*a(n-4)=0. - R. J. Mathar, Jul 02 2018

Example

a(2)=3 because we have UUUDDUUDDD, UUUDUDUDDD and UUUUUDDDDD, where U=(1,1), D=(1,-1).

Maple

G:=(1-z-2*z^2-2*z^3-sqrt(1-3*z^2-2*z))/2/z^4/(2+2*z+z^2): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..30);

Mathematica

CoefficientList[Series[(1-x-2*x^2-2*x^3-Sqrt[1-3*x^2-2*x])/2/x^4 /(2+2*x+x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Prog

(PARI) x='x+('x^50); Vec((1-x-2*x^2-2*x^3-sqrt(1-3*x^2-2*x))/(2*x^4*(2+2*x+x^2))) \\ G. C. Greubel, Mar 17 2017

Crossrefs

Cf. A114588, A005043.

Sequence in context: A134184 A142975 A211277 * A192908 A078679 A025577

Adjacent sequences: A114586 A114587 A114588 * A114590 A114591 A114592

Keyword

nonn

Author

Emeric Deutsch, Dec 11 2005

Status

approved