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%%a Plain TeX file by Shalosh B. Ekhad (1 page)

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\bf
\centerline
{
You Don't Have To Be an Einstein to Figure Out that Sara Should (Asymptotically) Eat
}
\centerline
{
$\frac{n}{3}+ \frac{4}{27}+O(1/n)$  Dove Bars, 
}
\centerline
{
But you do Need to be an Ekhad (or any of its silicon brethren)
}
\centerline
{to Figure Out that She Should Eat (Asymptotically)}
\centerline
{
$\frac{n}{3}+\frac{4}{27}+\frac{1}{81n}+\frac{5}{729n^2}+\frac{23}{6561n^3}+ \frac{1}{6561n^4}-\frac{281}{59049n^5} -\frac{5855}{531441n^6}
-\frac{18691}{1594323n^7}+\frac{245947}{14348907n^8}+\frac{15502093}{129140163n^9}+\frac{106690105}{387420489n^{10}} +O(\frac{1}{n^{11}})$
}
\centerline
{Dove Bars}
\rm
\centerline{ {\it
Shalosh B. EKHAD }\footnote{$^1$}
{\eightrm  \raggedright
c/o D. Zeilberger,
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg  at math dot rutgers dot edu} (Subject: For Ekhad),
\hfill \break
Exclusively published in the Personal Journal of Shalosh B. Ekhad and D. Zeilberger
\hfill\break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/pj.html} .
Nov. 11, 2009.
Supported in part by the NSF.
}
}




{\bf First} download the Maple package: http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/AsyRec .
Then, go into maple, and type:

read AsyRec:

expand(Asy(SumTools[Hypergeometric][Zeilberger]((k*binomial(n+k,k)*binomial(n,n-2*k)+
k*binomial(n+k-1,k-1)*binomial(n,n-2*k+1))/binomial(2*n,n),n,k,N)[1],n,N,10)/3);

and you would get the answer in the title, 
that improves from $O(1/n)$ to $O(1/n^{11})$ the asymptotic formula,
derived by clever (human) arguments by D.M. Einstein, C.C. Heckman, and T.S. Norfolk
({\it Amer. Math. Monthly} {\bf 116} (2009), 831-835). Of course, with a few more seconds of computations,
you can get $O(1/n^{30})$ and beyond, but do you really care? Sara would get a terrible tummy ache way before
even the $O(1/n^{11})$ asymptotics will start to be useful (and probably even before the original,
humanly-derived, $O(1/n)$ asymptotics).




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