OFFSET
0,4
COMMENTS
Column 0 of A109077.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 2*(1 -z +2*z^2 +(1-z)*q)/((1-2*z+q)*(1+2*z^2+q)), where q = sqrt(1-4*z^2).
a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence 4*(n+1)*a(n) +2*(-n-3)*a(n-1) +2*(-7*n+11)*a(n-2) +(7*n-27)*a(n-3) +2*(-4*n+5)*a(n-4) +4*(n-3)*a(n-5)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(4)=4 because we have uudduudd, uudududd, uuududdd and uuuudddd, where u=(1,1), d=(1,-1).
MAPLE
g:=2*(1-z-z*sqrt(1-4*z^2)+2*z^2+sqrt(1-4*z^2))/(1+sqrt(1-4*z^2)-2*z)/(1+sqrt(1-4*z^2)+2*z^2): gser:=series(g, z=0, 39): 1, seq(coeff(gser, z^n), n=1..36);
MATHEMATICA
CoefficientList[Series[2*(1-x-x*Sqrt[1-4*x^2]+2*x^2+Sqrt[1-4*x^2])/(1+ Sqrt[1-4*x^2]-2*x)/(1+Sqrt[1-4*x^2]+2*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) my(x='x+O('x^40)); Vec(2*(1-x-x*sqrt(1-4*x^2)+2*x^2 +sqrt(1-4*x^2))/(1+sqrt(1-4*x^2)-2*x)/(1+sqrt(1-4*x^2)+2*x^2)) \\ G. C. Greubel, Mar 16 2017
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2*(1-x-x*Sqrt(1-4*x^2)+2*x^2 +Sqrt(1-4*x^2))/(1+Sqrt(1-4*x^2)-2*x)/(1+Sqrt(1-4*x^2)+2*x^2) )); // G. C. Greubel, Apr 29 2019
(Sage) (2*(1-x-x*sqrt(1-4*x^2)+2*x^2 +sqrt(1-4*x^2))/(1+sqrt(1-4*x^2)-2*x)/(1+sqrt(1-4*x^2)+2*x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 17 2005
STATUS
approved