A227571 -id:A227571 - cp's OEIS frontend

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

Wolfdieter Lang, Jul 20 2006

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.

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

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.

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).

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

[Numerator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023

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 *)

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

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

A227570 Numerators of rationals with e.g.f. D(3,x), a Debye function.

Original entry on oeis.org

1, -3, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 16 2013

The denominators are given in A227571.
For general remarks on the e.g.f.s D(n,x), the Debye function with index n = 1, 2, 3, ... see the W. Lang link under A120080.
D(3,x) := (3/x^3)*int(t^3/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(3,n) = 3*B(n)/(n+3), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).
See the Abramowitz-Stegun link for the integral appearing in
  D(3,x) and a series expansion valid for |x| < 2*Pi.
Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Aug 13 2013
Differs from these sequences at n = 1292, 2624, 2770, 2778.... - Andrey Zabolotskiy, Dec 08 2023

			The rationals r(3,n), n=0..15 are: 1, -3/8, 1/10, 0, -1/70, 0, 1/126, 0, -1/110, 0, 5/286, 0, -691/13650, 0, 7/34, 0.

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.

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.

Cf. A227571, A227573, A027641/A027642, A120080/A120081 (D(3,x) as o.g.f.).

A227570[n_]:=Numerator[3BernoulliB[n]/(n+3)];
Array[A227570,50,0] (* Paolo Xausa, Dec 08 2023 *)

print([(bernoulli(n)*3/(n+3)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

a(n) = numerator(3*B(n)/(n+3)), n >= 0, with the Bernoulli numbers B(n).

cp's OEIS Frontend

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

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