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Numerators of expansion for Debye function for n=2: D(2,x).
5

%I #28 May 03 2023 09:13:21

%S 1,-1,1,0,-1,0,1,0,-1,0,1,0,-691,0,1,0,-3617,0,43867,0,-174611,0,

%T 77683,0,-236364091,0,657931,0,-3392780147,0,1723168255201,0,

%U -7709321041217,0,151628697551,0,-26315271553053477373

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

%C Denominators are found under A120085.

%C This sequence appears to coincide with A120082.

%H G. C. Greubel, <a href="/A120084/b120084.txt">Table of n, a(n) for n = 0..500</a>

%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 [alternative scanned copy].

%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=1, with a factor (x^2)/2 extracted.

%H Wolfdieter Lang, <a href="/A120084/a120084.txt">Rationals r(n)</a>.

%F a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - 2*x/(2*(2+1)) + Sum_{k >= 0} ((B(2*k)/((k+1)*(2*k)!))*x^(2*k) ), |x|<2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

%F 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 Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960, ...].

%t 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 Table[Numerator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* _G. C. Greubel_, May 02 2023 *)

%o (Magma) [Numerator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // _G. C. Greubel_, May 02 2023

%o (SageMath) [numerator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # _G. C. Greubel_, May 02 2023

%Y Cf. A000367, A002445, A120080, A120081, A120082, A120083, A120085, A120086, A120087.

%K sign,frac

%O 0,13

%A _Wolfdieter Lang_, Jul 20 2006