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A120082
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Numerators of expansion for Debye function for n=1: D(1,x).
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13
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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, 0, 154210205991661, 0, -261082718496449122051
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OFFSET
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0,13
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COMMENTS
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Denominators are found under A120083.
Let zeta(n) denote the Riemann zeta function and let [n even] be 1 if n is even, 0 otherwise, A059841. Further let n$ denote the swinging factorial of n (A056040). The swinging Bernoulli numbers are b_n = 2 zeta(n) n$ (2 Pi)^(-n) [n even] for n >= 2 and additionally b_0 = 1 and b_1 = 1/2, see A182918. a(n) are the numerators of b_n times a sign factor which is -1 if n=1 and (-1)^floor((n-1)/2)) otherwise. - Peter Luschny, Feb 03 2011
For n > 0 these are the numerators of the divided Bernoulli numbers, a(n) = B_n/n. - Peter Luschny, Jul 14 2013
D(1,x) = (1/x)*integral_{t=0..x} t/(exp(t)-1) dt (note the factor 1/x compared to the Abramowitz-Stegun link). This is the e.g.f. for {Bernoulli(n}/(n+1)}_{n>=0}. See A027641(n)/A227540(n). Thanks to Peter Luschny for asking me to revisit this sequence. - Wolfdieter Lang, Jul 15 2013
Also numerators of coefficients in expansion of x/(exp(x)-1). See A227829 for denominators. - N. J. A. Sloane, Aug 01 2013
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge University Press, p. 30, 2003.
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 23.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..628
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 extracted.
A. Adelberg, S. Hong and W. Ren, Bounds of divided universal Bernoulli numbers and universal Kummer congruences, Proc. of the American Mathematical Society, volume 136, number 1, January 2008, pages 61-71.
B. C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.
Wolfdieter Lang, Rationals r(n).
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FORMULA
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a(n) = numerator(r(n)), with r(n) = [x^n] (1 - x/4 + Sum_{k>=0} (B(2*k)/((2*k+1)*(2*k)!))*x^(2*k)), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).
a(n) = numerator(B(n)/n), n >= 1. See the P. Luschny comment and programs.
a(n) = numerator(B(n)/(n+1)!), n >= 0. See the above comment on the e.g.f. D(1,x). - Wolfdieter Lang, Jul 15 2013
Recurrence: R(0) = 1 and R(n) = -Sum_{k=0..n-1} R(k)/(n-k+1)! for n >= 1. Then a(n) = numerator(R(n)) and n!*R(n) = B(n) (Bernoulli numbers). - Peter Luschny, Jul 30 2015
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EXAMPLE
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Rationals r(n): [1, -1/4, 1/36, 0, -1/3600, 0, 1/211680, 0, -1/10886400, ...].
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MAPLE
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A120082 := proc(n) local b; if n = 0 then b := 1 ; elif n = 1 then b := -1/4 ; elif type(n, 'odd') then b := 0; else b := bernoulli(n)/(n+1)! ; fi; numer(b) ; end: # R. J. Mathar, Sep 03 2009
A120082 := proc(n) local swfact;
swfact := n -> n!/iquo(n, 2)!^2;
if n=0 then 1 elif n=1 then 1/2 else
if n mod 2 = 1 then 0
else 2*Zeta(n)*swfact(n)/(2*Pi)^n fi
fi;
`if`(n=1, -1, (-1)^iquo(n-1, 2))*numer(%) end:
seq(A120082(i), i=0..45); # Peter Luschny, Feb 03 2011
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MATHEMATICA
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swfact[n_] := n!/Floor[n/2]!^2; a[n_] := 2*Zeta[n]*swfact[n]/(2*Pi)^n*If[Mod[n, 4] == 0, -1, 1]; a[0] = 1; a[1] = -1; a[_?OddQ] = 0; Table[a[n], {n, 0, 45}] // Numerator (* Jean-François Alcover, Aug 09 2012, after Peter Luschny *)
Join[{1}, Rest @ CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 37}], 1/n]] // Numerator (* Jean-François Alcover, Mar 28 2013, after S. R. Finch *)
Join[{1}, Table[Numerator[BernoulliB[n]/n], {n, 45}]] (* Peter Luschny, Jul 14 2013 *)
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PROG
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(Sage)
def A120082(n): return (bernoulli(n)/n).numerator() if n > 0 else 1
[A120082(n) for n in (0..45)] # Peter Luschny, Jul 14 2013
(Sage)
@cached_function
def R(n): return -sum(R(k)/factorial(n-k+1) for k in (0..n-1)) if n>0 else 1
print([R(n).numerator() for n in (0..45)]) # Peter Luschny, Jul 30 2015
(PARI) a(n) = if (n==0, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015
(MAGMA) [1] cat [Numerator(Bernoulli(n)/(n)): n in [1..45]]; // G. C. Greubel, Sep 19 2019
(GAP) Concatenation([1], List([1..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019
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CROSSREFS
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Cf. A060054, A060055, A120083, A182918, A227540, A227829.
Sequence in context: A214335 A060054 A120084 * A249699 A141588 A281331
Adjacent sequences: A120079 A120080 A120081 * A120083 A120084 A120085
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KEYWORD
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sign,frac
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AUTHOR
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Wolfdieter Lang, Jul 20 2006
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EXTENSIONS
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Terms a(37) onward added by G. C. Greubel, Sep 19 2019
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STATUS
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approved
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