A120080 - OEIS

%I #24 May 01 2023 10:03:31

%S 1,-3,1,0,-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 original Debye function D(3,x).

%C Denominators are given in A120081.

%C See the W. Lang link below for more details on the general case D(n,x), n= 1, 2, ... D(3,x) is the e.g.f. of the rational sequence {3*B(n)/(n+3)}, n >= 0. See A227570/A227571.

%D L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band V: Statistische Physik, Akademie Verlag, Leipzig, p. 195, equ. (63.5) and footnote 1 on p. 197.

%H G. C. Greubel, <a href="/A120080/b120080.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=3, with a factor (x^3)/3 extracted.

%H Wolfdieter Lang, <a href="/A120080/a120080.pdf"> Rationals r(n), and general remarks on the e.g.f. D(n,x)</a>.

%F D(x) = D(3,x) := (3/x^3)*Integral_{0..x} (t^3/(exp(t)-1) dt.

%F a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - 3*x/8 + Sum_{k >= 0} ((B(2*k)/((2*k+3)*(2*k)!))*x^(2*k) ) (in lowest terms), |x| < 2*pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

%F a(n) = numerator(3*B(n)/((n+3)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). See the comment on the e.g.f. D(3,x) above. - _Wolfdieter Lang_, Jul 16 2013

%e Rationals r(n): [1, -3/8, 1/20, 0, 1/1680, 0, 1/90720, 0, ...].

%t max = 39; Numerator[CoefficientList[Integrate[Normal[Series[(3*(t^3/(Exp[t] - 1)))/x^3, {t, 0, max}]], {t, 0, x}], x]] (* _Jean-François Alcover_, Oct 04 2011 *)

%t Table[Numerator[3*BernoulliB[n]/((n+3)*n!)], {n,0,50}] (* _G. C. Greubel_, May 01 2023 *)

%o (Magma) [Numerator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // _G. C. Greubel_, May 01 2023

%o (SageMath)

%o def A120080(n): return numerator(3*bernoulli(n)/((n+3)*factorial(n)))

%o [A120080(n) for n in range(51)] # _G. C. Greubel_, May 01 2023

%Y Cf. A000367, A002445, A027641, A027642, A120081, A227570, A227571.

%K sign,frac

%O 0,2

%A _Wolfdieter Lang_, Jul 20 2006