This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120085 #22 Jan 25 2025 08:26:50
%S A120085 1,3,24,1,2160,1,120960,1,6048000,1,287400960,1,9153720576000,1,
%T A120085 597793996800,1,96035605585920000,1,51090942171709440000,1,
%U A120085 8831434289681203200000,1,169213200472701665280000,1,22019713777512667702886400000,1,2605883287279605645312000000
%N A120085 Denominators of expansion for Debye function for n=2: D(2,x).
%C A120085 Numerators are found under A120084.
%C A120085 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 A120085 See the Abramowitz-Stegun link for the integral and an expansion. - _Wolfdieter Lang_, Jul 16 2013
%H A120085 G. C. Greubel, <a href="/A120085/b120085.txt">Table of n, a(n) for n = 0..445</a>
%H A120085 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 A120085 a(n) = denominator(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 A120085 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 A120085 Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960,...].
%t A120085 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 A120085 Table[Denominator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* _G. C. Greubel_, May 02 2023 *)
%o A120085 (Magma) [Denominator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // _G. C. Greubel_, May 02 2023
%o A120085 (SageMath) [denominator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # _G. C. Greubel_, May 02 2023
%Y A120085 Cf. A000367/A002445, A027641/A027642, A120080/A120081 (D(3,x) expansion), A120082/A120083 (D(1,x) expansion), A120084, A120086, A120087.
%K A120085 nonn,easy,frac
%O A120085 0,2
%A A120085 _Wolfdieter Lang_, Jul 20 2006