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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 11 2005
STATUS
approved