

A321940


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


2



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

0,2


COMMENTS

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/(1x))*E1(1/(1x)), where E1(x) is an exponential integral. See Lemmas 12 and Theorem 5 of the preprint by Brent et al. (2018).


REFERENCES

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


LINKS

Table of n, a(n) for n=0..11.
Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
N. M. Temme, Remarks on Slater's asymptotic expansions of Kummer functions for large values of the aparameter, Adv. Dyn. Syst. Appl., 8 (2013), 365377.


FORMULA

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


EXAMPLE

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


CROSSREFS

The numerators are A321939. The formula in Theorem 5 of Brent et al. (2018) uses A321937(n)/A321938(n).
Sequence in context: A174755 A269419 A299422 * A222846 A265666 A114721
Adjacent sequences: A321937 A321938 A321939 * A321941 A321942 A321943


KEYWORD

nonn,frac


AUTHOR

Richard P. Brent, Dec 08 2018


STATUS

approved



