A321937 -id:A321937 - cp's OEIS frontend

A321938 Denominators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).

1, 12, 288, 51840, 2488320, 209018880, 75246796800, 180592312320, 86684309913600, 73557828698112000, 86504006548979712000, 13494625021640835072000, 9716130015581401251840000, 23318712037395363004416000, 559649088897488712105984000

Offset: 0

Richard P. Brent, Nov 22 2018

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).

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

			For n=0..3 the Maclaurin coefficients are 1, -1/12, 1/288, 67/61840.

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

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 a-parameter, Adv. Dyn. Syst. Appl., 8 (2013), 365-377.

Numerators are A321937.

A321938List := proc(len) local mu, ser;
mu  := h -> sum(bernoulli(2*k)/(2*k)!*h^(2*k-1), k=1..infinity);
ser := series(exp(mu(h)), h, len+2): seq(denom(coeff(ser,h,n)), n=0..len) end:
A321938List(14); # Peter Luschny, Dec 05 2018

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

x='x+O('x^25); apply(denominator, Vec(exp(1/x - 1/(exp(x)-1) - 1/2)))  \\ Joerg Arndt, Dec 05 2018

A321939 Numerators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).

Original entry on oeis.org

1, -5, -479, -15313, 710401, -3532731539, -1439747442109, -34886932972781, -171887027703456763, -6317295244143234168127, -2059266220658860906379923, -16155159358654324183625719723, -125609753430605939189919003924509

Offset: 0

Richard P. Brent, Dec 05 2018

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).

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

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

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 a-parameter, Adv. Dyn. Syst. Appl., 8 (2013), 365-377.

The denominators are A321940. The formula for A321939(n)/A321940(n) in Theorem 5 of Brent et al. (2018) uses A321937(n)/A321938(n). The sequence A321941 can be defined using A321939 and A321940.

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

A321940 Denominators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).

Original entry on oeis.org

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

Offset: 0

Richard P. Brent, Dec 08 2018

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).

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

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

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 a-parameter, Adv. Dyn. Syst. Appl., 8 (2013), 365-377.

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

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

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A321938 Denominators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).

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A321939 Numerators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).

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A321940 Denominators in the asymptotic expansion of the Maclaurin coefficients of exp(x/(1-x)).

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula