A120080 Numerators of expansion of original Debye function D(3,x).

1, -3, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373

Offset

0,2

Comments

Denominators are given in A120081.

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.

References

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 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.

Wolfdieter Lang, Rationals r(n), and general remarks on the e.g.f. D(n,x).

Formula

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

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).

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

Example

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

Mathematica

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 *)
Table[Numerator[3*BernoulliB[n]/((n+3)*n!)], {n, 0, 50}] (* G. C. Greubel, May 01 2023 *)

Prog

(Magma) [Numerator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023
(SageMath)
def A120080(n): return numerator(3*bernoulli(n)/((n+3)*factorial(n)))
[A120080(n) for n in range(51)] # G. C. Greubel, May 01 2023

Crossrefs

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

Sequence in context: A307766 A025443 A377128 * A227570 A352269 A111700

Adjacent sequences: A120077 A120078 A120079 * A120081 A120082 A120083

Keyword

sign,frac

Author

Wolfdieter Lang, Jul 20 2006

Status

approved