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%I #23 Feb 12 2021 12:12:11
%S 1,-1,1,67,-283,-5911,269891,114551,-9390523,-1021798901,273468378049,
%T 3918564638257,-872697935308349,-131115162268691,1397912875942181,
%U 2172284899403876321,-3926446823184958835813,-284746035618826337921,286113629384558337084185927
%N Numerators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).
%C The Maclaurin coefficients arise in a theorem of Slater (1960) on asymptotic expansions of confluent hypergeometric functions, see Sec. 3.1 of the paper by Temme (2013), and Theorem 5 of the preprint by Brent et al. (2018).
%D L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
%H 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 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.
%e For n=0..3 the Maclaurin coefficients are 1, -1/12, 1/288, 67/61840.
%p A321937List := proc(len) local mu, ser;
%p mu := h -> sum(bernoulli(2*k)/(2*k)!*h^(2*k-1), k=1..infinity);
%p ser := series(exp(mu(-h)), h, len+2): seq(numer(coeff(ser,h,n)), n=0..len) end:
%p A321937List(18); # _Peter Luschny_, Dec 05 2018
%t Exp[1/x - 1/(Exp[x]-1) - 1/2] + O[x]^20 // CoefficientList[#, x]& // Numerator (* _Jean-François Alcover_, Jan 21 2019 *)
%o (PARI) x='x+O('x^25); apply(numerator ,Vec(exp(1/x - 1/(exp(x)-1) - 1/2))) \\ _Joerg Arndt_, Dec 05 2018
%Y Denominators are A321938.
%K sign,frac
%O 0,4
%A _Richard P. Brent_, Nov 22 2018
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