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A120084 Numerators of expansion for Debye function for n=2: D(2,x). 5
1, -1, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,13
COMMENTS
Denominators are found under A120085.
This sequence appears to coincide with A120082.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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=1, with a factor (x^2)/2 extracted.
Wolfdieter Lang, Rationals r(n).
FORMULA
a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - 2*x/(2*(2+1)) + 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) = numerator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) in A120085. - Wolfdieter Lang, Dec 03 2022
EXAMPLE
Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960, ...].
MATHEMATICA
max = 38; Numerator[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[Numerator[2*(n+1)*BernoulliB[n]/(n+2)!], {n, 0, 50}] (* G. C. Greubel, May 02 2023 *)
PROG
(Magma) [Numerator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023
(SageMath) [numerator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023
CROSSREFS
Sequence in context: A263114 A214335 A060054 * A120082 A358625 A249699
KEYWORD
sign,frac
AUTHOR
Wolfdieter Lang, Jul 20 2006
STATUS
approved

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Last modified June 20 17:22 EDT 2024. Contains 373530 sequences. (Running on oeis4.)