A Maple One-Liner that is a MUCH Better ANSWER than George Andrews’ “Explicit” Formula for the Rademacher Coefficients

Shalosh B. EKHAD ^{1}

Drew Sills and my beloved master, Doron Zeilberger, have just posted

(http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/hans.html) a beautiful ar-

ticle 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 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 (Ramanujan J. 7 (2003) 385-400

; http://www.springerlink.com/content/q7g361w42231r247/ ), and proved (Theorem 1., p.

388) an 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)

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: seq(H(i),i=1..70); . In 51 seconds, one gets that H(195) is 77.796670050 … (a far cry from −.292927 …, conjectured by Rademacher to be the limit of H(n) as n → ∞).

Using more clever ideas, Sills and Zeilberger get many more terms, and were able to make a 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 formula is an algorithm, and Andrews’ “formula” is a very lousy algorithm. See Herb Wilf’s classic “What is an

answer” (http://www.jstor.org/pss/2321713) for other examples.

1 c/o zeilberg at math dot rutgers dot edu.

Exclusively published in: http://www.math.rutgers.edu/~zeilberg/pj.html . Oct. 23, 2011.