%I #21 Dec 08 2023 23:36:56
%S 1,-3,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,
%T 854513,0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,
%U -7709321041217,0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N Numerators of rationals with e.g.f. D(3,x), a Debye function.
%C The denominators are given in A227571.
%C 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.
%C 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).
%C See the Abramowitz-Stegun link for the integral appearing in
%C D(3,x) and a series expansion valid for |x| < 2*Pi.
%C Initially coincides with A176327, A164555 and A027641 for n <> 1. - _R. J. Mathar_, Aug 13 2013
%C Differs from these sequences at n = 1292, 2624, 2770, 2778.... - _Andrey Zabolotskiy_, Dec 08 2023
%D 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.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, 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.
%F a(n) = numerator(3*B(n)/(n+3)), n >= 0, with the Bernoulli numbers B(n).
%e 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.
%t A227570[n_]:=Numerator[3BernoulliB[n]/(n+3)];
%t Array[A227570,50,0] (* _Paolo Xausa_, Dec 08 2023 *)
%o (Sage)
%o print([(bernoulli(n)*3/(n+3)).numerator() for n in range(30)]) # _Andrey Zabolotskiy_, Dec 08 2023
%Y Cf. A227571, A227573, A027641/A027642, A120080/A120081 (D(3,x) as o.g.f.).
%K sign,easy,frac
%O 0,2
%A _Wolfdieter Lang_, Jul 16 2013