%I #19 May 02 2023 05:10:51
%S 1,3,24,1,2160,1,120960,1,6048000,1,287400960,1,9153720576000,1,
%T 597793996800,1,96035605585920000,1,51090942171709440000,1,
%U 8831434289681203200000,1,169213200472701665280000,1
%N Denominators of expansion for Debye function for n=2: D(2,x).
%C Numerators are found under A120084.
%C D(2,x) := (2/x^2)*Integral_{0..x} (t^2/(exp(t)-1) dt is the e.g.f. of 2*B(n)/(n+2), n>=0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). Proof by using the e.g.f. for {k*B(k-1)} (with 0 for k=0) and integrating termwise (allowed for |x| <= r < rho with small enough rho).
%C See the Abramowitz-Stegun link for the integral and an expansion. - _Wolfdieter Lang_, Jul 16 2013
%H G. C. Greubel, <a href="/A120085/b120085.txt">Table of n, a(n) for n = 0..445</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, pp. 998, equ. 27.1.1 for n=2, multiplied by 2/x^2.
%F a(n) = denominator(r(n)), with r(n):=[x^n](1 - x/3 + 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) = denominator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) above. - _Wolfdieter Lang_, Jul 16 2013
%e Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960,...].
%t max = 25; Denominator[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[Denominator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* _G. C. Greubel_, May 02 2023 *)
%o (Magma) [Denominator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // _G. C. Greubel_, May 02 2023
%o (SageMath) [denominator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # _G. C. Greubel_, May 02 2023
%Y Cf. A000367/A002445, A027641/A027642, A120080/A120081 (D(3,x) expansion), A120082/A120083 (D(1,x) expansion), A120084, A120086, A120087.
%K nonn,easy,frac
%O 0,2
%A _Wolfdieter Lang_, Jul 20 2006