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

If r(n) = A067764(n)/A067653(n) then r(n)/(exp(2*sqrt(n))/(2*n^(3/4)*sqrt(piPi*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.

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).

A formula is given in Theorem 5, and a recurrence in Lemma 7, of Brent et al . (2018).

The numerators are A321939. The formula in Theorem 5 of Brent et al . (2018) uses A321937(n)/A321938(n).

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1, 48, 4608, 3317760, 127401984, 214035333120, 308210879692800, 2958824445050880, 5680942934497689600, 134979204123665104896000, 18141205034220590098022400, 56600559706768241105829888000

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1, -5, -479, -15313, 710401, -3532731539, -1439747442109, -34886932972781

1, 48, 4608, 3317760, 127401984, 214035333120, 308210879692800, 2958824445050880

allocated for Richard P. Brent

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

1, -5, -479, -15313, 710401, -3532731539, -1439747442109, -34886932972781

0,2

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.

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).

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).

L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.

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.

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.

A formula is given in Theorem 5, and a recurrence in Lemma 7, of Brent et al (2018).

The asymptotic expansion is 1 - 5*h/48 - 479*h^2/4608 - 15313*h^3/3317760 + ..., where h = 1/sqrt(n).

The numerators are A321939. The formula in Theorem 5 of Brent et al (2018) uses A321937(n)/A321938(n).

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Richard P. Brent, Dec 08 2018

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allocated for Richard P. Brent

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