A120082 Numerators of expansion for Debye function for n=1: D(1,x).

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

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• sign
• frac

Comments

Denominators are found under A120083.

D(1,x) = (1/x)*integral_{t=0..x} t/(exp(t)-1) dt (note the factor x on the r.h.s. of the Abramowitz-Stegun link). This is the e.g.f. for {Bernoulli(n)/(n+1)}A027641(n)/A227540(n).%20Thanks%20to%20_Peter%20Luschny">{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 A227830 for denominators. - N. J. A. Sloane, Aug 01 2013

Examples

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

References

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

Links

• Peter Luschny, Table of n, a(n) for n = 0..500
• 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.
• Wolfdieter Lang, Rationals r(n).

Crossrefs

Cf. A060054, A060055, A120083, A182918, A227540, A227830, A358625.

Programs

Magma

[Numerator(Bernoulli(n)/Factorial(n+1)): n in [0..50]]; // G. C. Greubel, May 01 2023

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
gf := (1 - x/4 + sum((bernoulli(2*k)/((2*k+1)*(2*k)!))*x^(2*k), k=0..infinity)):
a := proc(n) local ser; if n = 0 then return 1 fi; ser := series(gf, x, n+2):
numer(coeff(ser, x, n)) end: seq(a(n), n = 0..40); # Peter Luschny, Dec 02 2022

Mathematica

Table[Numerator[BernoulliB[n]/((n+1)!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

SageMath

def A120082(n): return numerator(bernoulli(n)/factorial(n+1))
[A120082(n) for n in range(51)] # G. C. Greubel, May 01 2023

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+1)!), n >= 0. See the above comment on the e.g.f. D(1,x). - Wolfdieter Lang, Jul 15 2013

Apart from the sign of a(1) this sequence differs from A358625 for the first time at n = 68. - Peter Luschny, Dec 02 2022

Extensions

Edited after Andrey Zabolotskiy noticed an inconsistency by Peter Luschny, Dec 02 2022