A227570 -id:A227570 - 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

A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.

Original entry on oeis.org

1, -2, 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 17 2013

The denominators are given in A227574.
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(4,x) := (4/x^4)*int(t^4/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(4,n) = 4*B(n)/(n+4), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).
See the Abramowitz-Stegun reference for the integral appearing in
  D(4,x) and a series expansion valid for |x| < 2*Pi.
Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Jul 19 2013
Differs from these sequences for n = 1328, 2660, 2828, 2880... - Andrey Zabolotskiy, Dec 08 2023

			The rationals r(4,n), n=0..15 are: 1, -2/5, 1/9, 0, -1/60, 0, 1/105, 0, -1/90, 0, 5/231, 0, -691/10920, 0, 7/27, 0.

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=4, with a factor (x^4)/4 extracted.

Cf. A227570, A227574, A027641/A027642, A120086/A120087 (D(4,x) as o.g.f.).

A227573[n_]:=Numerator[4BernoulliB[n]/(n+4)];
Array[A227573,50,0] (* Paolo Xausa, Dec 08 2023 *)

print([(bernoulli(n)*4/(n+4)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

a(n) = numerator(4*B(n)/(n+4)), n >= 0, with the Bernoulli numbers B(n).

A227571 Denominators of rationals with e.g.f. D(3,x), a Debye function.

Original entry on oeis.org

1, 8, 10, 1, 70, 1, 126, 1, 110, 1, 286, 1, 13650, 1, 34, 1, 3230, 1, 5586, 1, 2530, 1, 1150, 1, 24570, 1, 58, 1, 8990, 1, 157542, 1, 5950, 1, 74, 1, 24949470, 1, 82, 1, 193930, 1, 27090, 1, 10810, 1, 4606, 1, 788970, 1, 1166, 1, 29150, 1, 15162, 1

Offset: 0

Wolfdieter Lang, Jul 16 2013

See the comments, references and links under A227570.

			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.

Cf. A227570, A027641/A027642.

A227571[n_]:=Denominator[3BernoulliB[n]/(n+3)];
Array[A227571,100,0] (* Paolo Xausa, Dec 08 2023 *)

a(n) = denominator(3*B(n)/(n+3)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).

The e.g.f. of the rationals r(3,n) := 3*B(n)/(n+3) is D(3,x) = (3/x^3)*int(t^3/(exp(x) - 1), t=0..x).

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

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A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.

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A227571 Denominators of rationals with e.g.f. D(3,x), a Debye function.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula