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A120082 Numerators of expansion for Debye function for n=1: D(1,x). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

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

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge University Press, p. 30, 2003.

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 23.

LINKS

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).

FORMULA

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

EXAMPLE

Rationals r(n): [1, -1/4, 1/36, 0, -1/3600, 0, 1/211680, 0, -1/10886400, ...].

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A060054, A060055, A120083, A182918, A227540, A227829.

Sequence in context: A214335 A060054 A120084 * A249699 A141588 A281331

Adjacent sequences:  A120079 A120080 A120081 * A120083 A120084 A120085

KEYWORD

sign,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

EXTENSIONS

Terms a(37) onward added by G. C. Greubel, Sep 19 2019

STATUS

approved

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Last modified September 18 09:13 EDT 2020. Contains 337166 sequences. (Running on oeis4.)