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Denominators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).
3

%I #23 Feb 12 2021 12:11:55

%S 1,12,288,51840,2488320,209018880,75246796800,180592312320,

%T 86684309913600,73557828698112000,86504006548979712000,

%U 13494625021640835072000,9716130015581401251840000,23318712037395363004416000,559649088897488712105984000

%N Denominators 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).

%C The sequence is related to A001164 but differs from the 7th term.

%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 A321938List := 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(denom(coeff(ser,h,n)), n=0..len) end:

%p A321938List(14); # _Peter Luschny_, Dec 05 2018

%t Exp[1/x - 1/(Exp[x]-1) - 1/2] + O[x]^20 // CoefficientList[#, x]& // Denominator (* _Jean-François Alcover_, Jan 21 2019 *)

%o (PARI) x='x+O('x^25); apply(denominator, Vec(exp(1/x - 1/(exp(x)-1) - 1/2))) \\ _Joerg Arndt_, Dec 05 2018

%Y Numerators are A321937.

%K nonn,frac

%O 0,2

%A _Richard P. Brent_, Nov 22 2018