OFFSET
0,3
COMMENTS
Column 0 of A114499.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/(1 - z*C + z^3), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) ~ 4^(n+5)/(1089*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence +(n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-2) +2*(-2*n+1)*a(n-3) +(n+1)*a(n-5) +2*(-2*n+1)*a(n-6)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(4)=12 because among the 14 Dyck paths of semilength 4 only UDUUUDDD and UUUDDDUD contain UUUDDD starting at level 0.
MAPLE
C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..30);
MATHEMATICA
CoefficientList[Series[1/(1-x*(1-Sqrt[1-4*x])/2/x+x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) my(x='x+O('x^50)); Vec(2/(1+sqrt(1-4*x)+2*x^3)) \\ Jason Yuen, Sep 09 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 04 2005
STATUS
approved