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A120085 Denominators of expansion for Debye function for n=2: D(2,x). 4
1, 3, 24, 1, 2160, 1, 120960, 1, 6048000, 1, 287400960, 1, 9153720576000, 1, 597793996800, 1, 96035605585920000, 1, 51090942171709440000, 1, 8831434289681203200000, 1, 169213200472701665280000, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
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.
FORMULA
a(n) = denominator(r(n)), with r(n):=[x^n](1 - x/3 + Sum_{k >= 0} 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
EXAMPLE
Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960,...].
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 *)
PROG
(Magma) [Denominator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [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
CROSSREFS
Cf. A000367/A002445, A027641/A027642, A120080/A120081 (D(3,x) expansion), A120082/A120083 (D(1,x) expansion), A120084, A120086, A120087.
Sequence in context: A355960 A264929 A204578 * A062834 A047980 A084702
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jul 20 2006
STATUS
approved

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Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)