A227573 - OEIS

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

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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