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A321940 Denominators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).

%I A321940 #10 Dec 09 2018 03:24:55
%S A321940 1,48,4608,3317760,127401984,214035333120,308210879692800,
%T A321940 2958824445050880,5680942934497689600,134979204123665104896000,
%U A321940 18141205034220590098022400,56600559706768241105829888000
%N A321940 Denominators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).
%C A321940 If r(n) = A067764(n)/A067653(n) then r(n)/(exp(2*sqrt(n))/(2*n^(3/4)*sqrt(Pi*e))) has an asymptotic expansion in ascending powers of 1/sqrt(n) whose coefficients are rational numbers 1, -5/48, etc. The sequence gives the denominators of these rational numbers.
%C A321940 Another expression for r(n), n > 0, is r(n) = M(n+1,2,1)/e, where M(a,b,z) = 1F1(a;b;z) is a confluent hypergeometric function (Kummer function).
%C A321940 The same rational numbers, except for signs, occur in the asymptotic expansion of the Maclaurin coefficients of exp(1/(1-x))*E1(1/(1-x)), where E1(x) is an exponential integral. See Lemmas 1-2 and Theorem 5 of the preprint by Brent et al. (2018).
%D A321940 L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
%H A321940 Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%H A321940 N. M. Temme, <a href="http://campus.mst.edu/adsa/contents/v8n2p16.pdf">Remarks on Slater's asymptotic expansions of Kummer functions for large values of the a-parameter</a>, Adv. Dyn. Syst. Appl., 8 (2013), 365-377.
%F A321940 A formula is given in Theorem 5, and a recurrence in Lemma 7, of Brent et al. (2018).
%e A321940 The asymptotic expansion is 1 - 5*h/48 - 479*h^2/4608 - 15313*h^3/3317760 + ..., where h = 1/sqrt(n).
%Y A321940 The numerators are A321939. The formula in Theorem 5 of Brent et al. (2018) uses A321937(n)/A321938(n).
%K A321940 nonn,frac
%O A321940 0,2
%A A321940 _Richard P. Brent_, Dec 08 2018