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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).
2
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
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
Sequence in context: A134184 A142975 A211277 * A192908 A078679 A025577
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 11 2005
STATUS
approved