A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.

1, -2, 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 17 2013

Keywords

• sign
• easy
• frac
• changed

Comments

The denominators are given in A227574.

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(4,x) := (4/x^4)*int(t^4/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(4,n) = 4*B(n)/(n+4), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).

See the Abramowitz-Stegun reference for the integral appearing in

D(4,x) and a series expansion valid for |x| < 2*Pi.

Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Jul 19 2013

Differs from these sequences for n = 1328, 2660, 2828, 2880... - Andrey Zabolotskiy, Dec 08 2023

Examples

			The rationals r(4,n), n=0..15 are: 1, -2/5, 1/9, 0, -1/60, 0, 1/105, 0, -1/90, 0, 5/231, 0, -691/10920, 0, 7/27, 0.
		

References

• 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=4, with a factor (x^4)/4 extracted.

Crossrefs

Cf. A227570, A227574, A027641/A027642, A120086/A120087 (D(4,x) as o.g.f.).

Programs

Mathematica

A227573[n_]:=Numerator[4BernoulliB[n]/(n+4)];
Array[A227573,50,0] (* Paolo Xausa, Dec 08 2023 *)

Sage

print([(bernoulli(n)*4/(n+4)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

Formula

a(n) = numerator(4*B(n)/(n+4)), n >= 0, with the Bernoulli numbers B(n).