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

%I A227573 #21 Aug 25 2025 10:56:59
%S A227573 1,-2,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,
%T A227573 854513,0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,
%U A227573 -7709321041217,0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.
%C A227573 The denominators are given in A227574.
%C A227573 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 A227573 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).
%C A227573 See the Abramowitz-Stegun reference for the integral appearing in
%C A227573   D(4,x) and a series expansion valid for |x| < 2*Pi.
%C A227573 Initially coincides with A176327, A164555 and A027641 for n <> 1. - _R. J. Mathar_, Jul 19 2013
%C A227573 Differs from these sequences for n = 1328, 2660, 2828, 2880... - _Andrey Zabolotskiy_, Dec 08 2023
%D A227573 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.
%F A227573 a(n) = numerator(4*B(n)/(n+4)), n >= 0,  with the Bernoulli numbers B(n).
%e A227573 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.
%t A227573 A227573[n_]:=Numerator[4BernoulliB[n]/(n+4)];
%t A227573 Array[A227573,50,0] (* _Paolo Xausa_, Dec 08 2023 *)
%o A227573 (Sage)
%o A227573 print([(bernoulli(n)*4/(n+4)).numerator() for n in range(30)]) # _Andrey Zabolotskiy_, Dec 08 2023
%Y A227573 Cf. A227570, A227574, A027641/A027642, A120086/A120087 (D(4,x) as o.g.f.).
%K A227573 sign,easy,frac,changed
%O A227573 0,2
%A A227573 _Wolfdieter Lang_, Jul 17 2013