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A120084 Numerators of expansion for Debye function for n=2: D(2,x).

%I A120084 #34 Apr 01 2025 07:38:07
%S A120084 1,-1,1,0,-1,0,1,0,-1,0,1,0,-691,0,1,0,-3617,0,43867,0,-174611,0,
%T A120084 77683,0,-236364091,0,657931,0,-3392780147,0,1723168255201,0,
%U A120084 -7709321041217,0,151628697551,0,-26315271553053477373,0,154210205991661,0,-261082718496449122051
%N A120084 Numerators of expansion for Debye function for n=2: D(2,x).
%C A120084 Denominators are found under A120085.
%C A120084 This sequence appears to coincide with A120082.
%H A120084 G. C. Greubel, <a href="/A120084/b120084.txt">Table of n, a(n) for n = 0..500</a>
%H A120084 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=1, with a factor (x^2)/2 extracted.
%H A120084 Wolfdieter Lang, <a href="/A120084/a120084.txt">Rationals r(n)</a>.
%F A120084 a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - x/3 + Sum_{k >= 1} (B(2*k)/((k+1)*(2*k)!))*x^(2*k) ), |x|<2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).
%F A120084 a(n) = numerator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) in A120085. - _Wolfdieter Lang_, Dec 03 2022
%e A120084 Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960, ...].
%t A120084 max = 38; Numerator[CoefficientList[Integrate[Normal[Series[(2*(t^2/(Exp[t]-1)))/x^2, {t, 0, max}]], {t, 0, x}], x]] (* _Jean-François Alcover_, Oct 04 2011 *)
%t A120084 Table[Numerator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* _G. C. Greubel_, May 02 2023 *)
%o A120084 (Magma) [Numerator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // _G. C. Greubel_, May 02 2023
%o A120084 (SageMath) [numerator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # _G. C. Greubel_, May 02 2023
%Y A120084 Cf. A000367, A002445, A120080, A120081, A120082, A120083, A120085, A120086, A120087.
%K A120084 sign,frac
%O A120084 0,13
%A A120084 _Wolfdieter Lang_, Jul 20 2006