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A321939
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Numerators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).
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3
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1, -5, -479, -15313, 710401, -3532731539, -1439747442109, -34886932972781, -171887027703456763, -6317295244143234168127, -2059266220658860906379923, -16155159358654324183625719723, -125609753430605939189919003924509
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OFFSET
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0,2
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COMMENTS
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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 numerators 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).
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REFERENCES
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L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
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LINKS
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FORMULA
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A formula is given in Theorem 5, and a recurrence in Lemma 7, of Brent et al. (2018).
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EXAMPLE
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The asymptotic expansion is 1 - 5*h/48 - 479*h^2/4608 - 15313*h^3/3317760 + ..., where h = 1/sqrt(n).
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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