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A227570 Numerators of rationals with e.g.f. D(3,x), a Debye function. 3
1, -3, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The denominators are given in A227571.

For general remarks on the e.g.f.s D(n,x), the Debye function with index n = 1, 2, 3, ... see the W. Lang link under A120080.

D(3,x) := (3/x^3)*int(t^3/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(3,n) = 3*B(n)/(n+3), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).

See the Abramowitz-Stegun link for the integral appearing in

  D(3,x) and a series expansion valid for |x| < 2*pi.

Appears to coincide with A176327, A164555 and A027641 for all n <> 1. - R. J. Mathar, Aug 13 2013

REFERENCES

L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band V: Statistische Physik, Akademie Verlag, Leipzig, p. 195, equ. (63.5), and footnote 1 on p. 197.

LINKS

Table of n, a(n) for n=0..40.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=3, with a factor (x^3)/3 extracted.

FORMULA

a(n) = numerator(3*B(n)/(n+3)), n >= 0,  with the Bernoulli numbers B(n).

EXAMPLE

The rationals r(3,n), n=0..15 are: 1, -3/8, 1/10, 0, -1/70, 0, 1/126, 0, -1/110, 0, 5/286, 0, -691/13650, 0, 7/34, 0.

CROSSREFS

Cf. A227571, A027641/A027642, A120080/A120081 (D(3,x) as o.g.f.).

Sequence in context: A307766 A025443 A120080 * A111700 A233316 A060096

Adjacent sequences:  A227567 A227568 A227569 * A227571 A227572 A227573

KEYWORD

sign,easy,frac

AUTHOR

Wolfdieter Lang, Jul 16 2013

STATUS

approved

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Last modified October 17 11:47 EDT 2019. Contains 328108 sequences. (Running on oeis4.)