A227570 Numerators of rationals with e.g.f. D(3,x), a Debye function.

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

Offset: 0

Author

Wolfdieter Lang, Jul 16 2013

Keywords

• sign
• easy
• frac

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.

Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Aug 13 2013

Differs from these sequences at n = 1292, 2624, 2770, 2778.... - Andrey Zabolotskiy, Dec 08 2023

Examples

			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.
		

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

• 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.

Crossrefs

Cf. A227571, A227573, A027641/A027642, A120080/A120081 (D(3,x) as o.g.f.).

Programs

Mathematica

A227570[n_]:=Numerator[3BernoulliB[n]/(n+3)];
Array[A227570,50,0] (* Paolo Xausa, Dec 08 2023 *)

Sage

print([(bernoulli(n)*3/(n+3)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

Formula

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