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

Author

Richard P. Brent, Nov 22 2018

Keywords

• nonn
• 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).

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

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

Numerators are A321937.

Programs

Maple

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

Mathematica

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

PARI

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