%I #26 May 02 2023 05:12:50
%S 1,-2,1,0,-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 of Debye function for n=4: D(4,x).
%C Denominators are found under A120087.
%C See the W. Lang link under A120080 for more details on the general case D(n,x), n= 1, 2, ... D(4,x) is the e.g.f. of the rational sequence {4*B(n)/(n+4)}, n >= 0. See A227573/A227574. - _Wolfdieter Lang_, Jul 17 2013
%H Vincenzo Librandi, <a href="/A120086/b120086.txt">Table of n, a(n) for n = 0..600</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^4)/4 extracted.
%H Wolfdieter Lang, <a href="/A120086/a120086.pdf">Rationals r(n).</a>
%F a(n) = numerator(r(n)), with r(n) = [x^n](1 - 4*x/(2*(4+1)) + 2*Sum_{k >= 0} (B(2*k)/((k+2)*(2*k)!))*x^(2*k) ), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).
%F a(n) = numerator(4*B(n)/((n+4)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). From D(4,x) read as o.g.f. - _Wolfdieter Lang_, Jul 17 2013
%e Rationals r(n): [1, -2/5, 1/18, 0, -1/1440, 0, 1/75600, 0, -1/3628800, 0, 1/167650560, 0, -691/5230697472000, ...].
%t r[n_]:= 4*BernoulliB[n]/((n+4)*n!); Table[r[n]//Numerator, {n,0,36}] (* _Jean-François Alcover_, Jun 21 2013 *)
%o (Magma) [Numerator(4*(n+1)*(n+2)*(n+3)*Bernoulli(n)/Factorial(n+4)): n in [0..50]]; // _G. C. Greubel_, May 02 2023
%o (SageMath) [numerator(4*(n+1)*(n+2)*(n+3)*bernoulli(n)/factorial(n+4)) for n in range(51)] # _G. C. Greubel_, May 02 2023
%Y Cf. A060054. [From _R. J. Mathar_, Aug 07 2008]
%Y Cf. A000367/A002445, A027641/A027642, A120097, A227573/A227574 (D(4,x) as e.g.f.). - _Wolfdieter Lang_, Jul 17 2013
%Y Cf. A120080, A120081, A120082, A120083, A120084, A120085, A120087 (denominators).
%K sign,frac
%O 0,2
%A _Wolfdieter Lang_, Jul 20 2006