A120081 Denominators of expansion for original Debye function (n=3).

1, 8, 20, 1, 1680, 1, 90720, 1, 4435200, 1, 207567360, 1, 6538371840000, 1, 423437414400, 1, 67580611338240000, 1, 35763659520196608000, 1, 6155242080686899200000, 1, 117509166994931712000000, 1, 15244417230585693025075200000, 1, 1799300365026394374144000000

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

Numerators are given in A120080.

See A120070 for the definition of the Debye function D(x)=D(3,x) and references and links.

Links

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

Crossrefs

Cf. A000367, A002445, A120080.

Programs

Magma

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

Mathematica

max = 26; Denominator[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[Denominator[3*BernoulliB[n]/((n+3)*n!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

SageMath

def A120081(n): return denominator(3*bernoulli(n)/((n+3)*factorial(n)))
[A120081(n) for n in range(51)] # G. C. Greubel, May 01 2023

Formula

a(n) = denominator(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) ), where B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator( 3*Bernoulli(n)/((n+3)*n!) ), n >= 0. - G. C. Greubel, May 01 2023