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%I #23 Jun 22 2021 12:41:44
%S -2,-6,-170,-9520,-874902,-118950678,-22370367448,-5550123527520
%N Rademacher's sequence C_{011}(N) times (2n)!, where C_{011}(N) is the coefficient of 1/(q-1) in the partial fraction decomposition of 1/((1-q)(1-q^2)...(1-q^N)).
%C Hans Rademacher conjectured that C_{011}(N) converge to -0.292927573960. This conjecture is false.
%C Named after the German-American mathematician Hans Adolph Rademacher (1892-1969). - _Amiram Eldar_, Jun 22 2021
%D Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, p. 302.
%H Andrew V. Sills and Doron Zeilberger, <a href="http://arxiv.org/abs/1110.4932">Rademacher's infinite partial fraction conjecture is (almost certainly) false</a>, arXiv:1110.4932v1 [math.NT], 2011.
%H Andrew V. Sills and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/hans.html">Rademacher's Infinite Partial Fraction Conjecture is (almost certainly) False</a>, Oct 21 2011; <a href="/A182523/a182523.pdf">Local copy, pdf file only, no active links</a>.
%H Andrew V. Sills and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/HANS">HANS (maple package)</a>; <a href="/A182523/a182523.txt">Local copy</a>.
%F See above article for an efficient recurrence.
%e For n=1, the coefficient of 1/(q-1) in the partial fraction decomposition of 1/(1-q) is -1, multiplied by 2! this gives -2.
%p See above link to HANS (maple package).
%K sign,more
%O 1,1
%A _Shalosh B. Ekhad_, May 03 2012
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