%HansGeorge.tex: A Maple One-Liner that is a MUCH Better ANSWER than George Andrews' ``Explicit'' Formula for the Rademacher Coefficients
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A Maple One-Liner that is a MUCH Better ANSWER than 
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George Andrews' ``Explicit'' Formula for the Rademacher Coefficients
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\centerline{ {\it Shalosh B. EKHAD }\footnote{$^1$}
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c/o {\eighttt zeilberg  at math dot rutgers dot edu}.
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Exclusively published in: {\eighttt http://www.math.rutgers.edu/\~{}zeilberg/pj.html} .
Oct. 23, 2011.
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Drew Sills and my beloved master, Doron Zeilberger, have just posted \hfill\break
({\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/hans.html})
a beautiful article refuting (``empirically'' but most convincingly) a
forty-year-old conjecture of the eminent number theorist Hans Rademacher
(that they were hoping that Andrews (Rademacher's academic {\it Benjamin}) would consider for the PNAS, 
but he declined, since ``empirical'' proofs do not count, and possibly also because it didn't
mention modular forms or Maass forms). George Andrews himself
attempted to take a first step towards proving this conjecture ({\it Ramanujan J.} {\bf 7} (2003) 385-400 ;
{\tt http://www.springerlink.com/content/q7g361w42231r247/ }),
and proved (Theorem 1., p. 388)  an {\it explicit} formula for the so-called $C_{011}(N+1)$.
OMG, with ``explicit'' formulas like this we don't need any implicit formulas.
It has $N!$ summands, each of which  takes (at least) exponential time to compute!!!

On the other hand, the  Maple one-liner code (Maple was readily available in 2003)

{\tt H:=N -> coeff(taylor(normal(y**N/mul(1 - (1+y)**i, i = 1 .. N)),y=0,N),y,N-1):}

enables one to get in 12 seconds the first 70 terms, by typing: {\tt seq(H(i),i=1..70);} .
In 51 seconds, one gets that {\tt H(195)} is $77.796670050\dots$ (a far cry from $-.292927\dots$,
conjectured by Rademacher to be the limit of $H(n)$ as $n \rightarrow \infty$).

Using more clever ideas, Sills and Zeilberger get many more terms, and were able to
make a {\it precise} conjecture about the sequence $\{ H(n)\}$ (and many related sequences),
but the above one-liner would have sufficed to show that the Rademacher conjecture is (almost certainly) false.


I don't know if the above Maple one-liner qualifies as an ``explicit formula'', but a {\it formula}
is an algorithm, and Andrews' ``formula'' is a {\it very lousy} algorithm. 
See Herb Wilf's classic ``{\it What is an answer}'' ({\tt http://www.jstor.org/pss/2321713})
for other examples.




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