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, 0, 154210205991661, 0, -261082718496449122051
Offset: 0
Examples
Rationals r(n): [1, -3/8, 1/20, 0, -1/1680, 0, 1/90720, 0, ...].
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, 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).
Programs
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Magma
[Numerator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023
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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 *) -
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
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 >= 1} (3*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
Comments