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

0,2

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 A227570, 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

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.

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A120086 A215030 A175816 * A085004 A095774 A266874

Adjacent sequences: A227570 A227571 A227572 * A227574 A227575 A227576

Keyword

sign,easy,frac

Author

Wolfdieter Lang, Jul 17 2013

Status

approved