A321937 Numerators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).

1, -1, 1, 67, -283, -5911, 269891, 114551, -9390523, -1021798901, 273468378049, 3918564638257, -872697935308349, -131115162268691, 1397912875942181, 2172284899403876321, -3926446823184958835813, -284746035618826337921, 286113629384558337084185927

Offset: 0

Author

Richard P. Brent, Nov 22 2018

Keywords

• sign
• frac

Comments

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).

Examples

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

References

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

Links

• 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.

Crossrefs

Denominators are A321938.

Programs

Maple

A321937List := 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(numer(coeff(ser,h,n)), n=0..len) end:
A321937List(18); # Peter Luschny, Dec 05 2018

Mathematica

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

PARI

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