You Don’t Have To Be an Einstein to Figure Out that Sara Should (Asymptotically) Eat

\frac{n}{3} + \frac{4}{27} + O(1/n) Dove Bars, But you do Need to be an Ekhad (or any of its silicon brethren) to Figure Out that She Should Eat (Asymptotically)

\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}} )

Dove Bars Shalosh B. EKHAD ^{1}

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 (Amer. Math. Monthly 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).

1 c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus,

110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu (Subject: For Ekhad), Exclusively published in the Personal Journal of Shalosh B. Ekhad and D. Zeilberger http://www.math.rutgers.edu/~zeilberg/pj.html . Nov. 11, 2009. Supported in part by the NSF.