A120085 Denominators of expansion for Debye function for n=2: D(2,x).

1, 3, 24, 1, 2160, 1, 120960, 1, 6048000, 1, 287400960, 1, 9153720576000, 1, 597793996800, 1, 96035605585920000, 1, 51090942171709440000, 1, 8831434289681203200000, 1, 169213200472701665280000, 1, 22019713777512667702886400000, 1, 2605883287279605645312000000

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

Numerators are found under A120084.

D(2,x) := (2/x^2)*Integral_{0..x} t^2/(exp(t)-1) dt is the e.g.f. of 2*B(n)/(n+2), n>=0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). Proof by using the e.g.f. for {k*B(k-1)} (with 0 for k=0) and integrating termwise (allowed for |x| <= r < rho with small enough rho).

See the Abramowitz-Stegun link for the integral and an expansion. - Wolfdieter Lang, Jul 16 2013

Examples

			Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960,...].
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• 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=2, multiplied by 2/x^2.

Crossrefs

Cf. A000367/A002445, A027641/A027642, A120080/A120081 (D(3,x) expansion), A120082/A120083 (D(1,x) expansion), A120084, A120086, A120087.

Programs

Magma

[Denominator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023

Mathematica

max = 25; Denominator[CoefficientList[Integrate[Normal[Series[(2*(t^2/(Exp[t]-1)))/x^2, {t, 0, max}]], {t, 0, x}], x]](* Jean-François Alcover, Oct 04 2011 *)
Table[Denominator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* G. C. Greubel, May 02 2023 *)

SageMath

[denominator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023

Formula

a(n) = denominator(r(n)), with r(n) = [x^n]( 1 - x/3 + Sum_{k >= 1} (B(2*k)/((k+1)*(2*k)!))*x^(2*k) ), |x|<2*pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) above. - Wolfdieter Lang, Jul 16 2013