The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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. 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 Adjacent sequences: A120082 A120083 A120084 * A120086 A120087 A120088 KEYWORD nonn,easy,frac AUTHOR Wolfdieter Lang, Jul 20 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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