A120082 -id:A120082 - cp's OEIS frontend

A227830 Denominators of coefficients in expansion of x/(exp(x)-1).

1, 2, 12, 1, 720, 1, 30240, 1, 1209600, 1, 47900160, 1, 1307674368000, 1, 74724249600, 1, 10670622842880000, 1, 5109094217170944000, 1, 802857662698291200000, 1, 14101100039391805440000, 1, 1693824136731743669452800000, 1, 186134520519971831808000000, 1, 37893265687455865519472640000000, 1, 759790291646040068357842010112000000, 1

Offset: 0

N. J. A. Sloane, Aug 01 2013

			1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600, 0, ...

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

For numerators see A120082.

Cf. A060054, A060055.

Denominator[ CoefficientList[ Series[x/(1 - E^-x), {x, 0, 26}], x]] (* Robert G. Wilson v, Dec 29 2016 *)

@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).denominator() for n in (0..31)]) # Peter Luschny, Jul 30 2015

Recurrence: R(0) = 1 and R(n) = - Sum_{k=0..n-1} R(k)/(n-k+1)! for n>=1. Then a(n) = denominator(R(n)). - Peter Luschny, Jul 30 2015

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Original entry on oeis.org

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

Peter Luschny, Dec 02 2022

The rational numbers r(n) = Bernoulli(n, 1) / n are called the 'divided Bernoulli numbers'. r(n) is a p-integer for all primes p if p - 1 does not divide n. This is sometimes called 'Adams's theorem' (Ireland and Rosen). The important Kummer congruences for the Bernoulli numbers (1851) are stated in terms of the r(n).

			Rationals: 1, 1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, ...
Note that a(68) = -4633713579924631067171126424027918014373353 but A120082(68) = -125235502160125163977598011460214000388469.

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]

Peter Luschny, Table of n, a(n) for n = 0..300
Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.
Cf. A164555 / A027642.
For the denominators cf. A006953, A036283, A053657, A202318.

Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019

[1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // G. C. Greubel, Sep 19 2019

A358625 := n -> ifelse(n = 0, 1, numer(bernoulli(n, 1) / n)):
seq(A358625(n), n = 0.. 40);
# Alternative:
egf := 1 + x + log(1 - exp(-x)) - log(x): ser := series(egf, x, 42):
seq(numer(n! * coeff(ser, x, n)), n = 0..40);

Join[{1, 1}, Table[Numerator[BernoulliB[n] / n], {n, 2, 45}]]

a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015

a(n) = numerator(n! * [x^n](1 + x + log(1 - exp(-x)) - log(x))).
a(n) = numerator(-zeta(1 - n)) for n >= 1.
a(n) = numerator(Euler(n-1, 1) / (2*(2^n - 1))) for n >= 1.
denominator(r(2*n)) = A006953(n) for n >= 1.
denominator(r(2*n)) / 2 = A036283(n) for n >= 1.
denominator(r(2*n)) / 12 = A202318(n) for n >= 1.
denominator(r(2*n)) = (1/2) * A053657(2*n+1) / A053657(2*n-1) for n >= 1.

A182918 Denominators of the swinging Bernoulli number b_n.

Original entry on oeis.org

1, 2, 6, 1, 120, 1, 1512, 1, 17280, 1, 190080, 1, 1415232000, 1, 21772800, 1, 829108224000, 1, 105082151731200, 1, 4345502515200000, 1, 19989311569920000, 1, 626378114550988800000, 1, 17896517558599680000, 1, 944578196742891110400000

Offset: 0

Peter Luschny, Feb 03 2011

Let zeta(n) denote the Riemann zeta function, B_n the Bernoulli numbers and let [n even] be 1 if n is even, 0 otherwise.
Then 2 zeta(n) [n even] = (2 Pi)^n | B_n | / n! for n >= 2.
Replacing in this formula the factorial of n by the swinging factorial of n (A056040) defines the 'swinging Bernoulli number' b_n.
Then 2 zeta(n) [n even] = (2 Pi)^n b_n / n$ for n >= 2.
Let additionally b_0 = 1 and b_1 = 1/2. The b_n are rational numbers like the Bernoulli numbers; unlike the Bernoulli numbers the swinging Bernoulli numbers are unsigned, bounded in the interval [0,1] and approach 0 for n -> infinity. The numerators of the swinging Bernoulli numbers b_n are abs(A120082(n)).

			1, 1/2, 1/6, 0, 1/120, 0, 1/1512, 0, 1/17280, 0, 1/190080, ..

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A120082.

swbern:= 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 end:
Abs_A120082 := n -> numer(swbern(n));
A182918 := n -> denom(swbern(n));
seq(A182918(i),i=0..20);

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[1] = 2; a[?OddQ] = 1; a[n] := 2*Zeta[n]*sf[n]/(2*Pi)^n // Denominator; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 26 2013 *)

A227540 Denominator of the rationals obtained from the e.g.f. D(1,x), a Debye function.

Original entry on oeis.org

1, 4, 18, 1, 150, 1, 294, 1, 270, 1, 726, 1, 35490, 1, 90, 1, 8670, 1, 15162, 1, 6930, 1, 3174, 1, 68250, 1, 162, 1, 25230, 1, 443982, 1, 16830, 1, 210, 1, 71010030, 1, 234, 1, 554730, 1, 77658, 1, 31050, 1, 13254, 1, 2274090, 1, 3366, 1, 84270, 1, 43890, 1

Offset: 0

Wolfdieter Lang, Jul 15 2013

The numerator sequence seems to be the one of the Bernoulli numbers A027641.

D(1,x) := (1/x)*int(t/(exp(t)-1),t=0..x) which is (1/x)times the Debye function of the Abramowitz-Stegun link for n=1, is the e.g.f. for {B(k)/(k+1)}, k=0..infinity, with the Bernoulli numbers B(k) = A027641(k)/A027642(k). This follows after using the e.g.f. t/(exp(t)-1) of {B(k)} and integrating term by term (allowed for |x| <= r < rho for some small enough rho).

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 an extra factor 1/x.

Cf. A027641/A027642 (Bernoulli), A120082/A120083 for the rationals B(n)/(n+1)!.

a(n) = denominator(B(n)/(n+1)) (in lowest terms), n >= 0. See the comment on the e.g.f. D(1,x) above.

A341908 Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.

Original entry on oeis.org

7, 7, 7, 5, 0, 4, 6, 3, 4, 1, 1, 2, 2, 4, 8, 2, 7, 6, 4, 1, 7, 5, 8, 6, 5, 4, 5, 4, 2, 5, 7, 1, 0, 5, 0, 7, 1, 9, 2, 4, 7, 7, 2, 9, 6, 2, 2, 9, 0, 0, 0, 8, 6, 9, 1, 7, 9, 4, 9, 4, 5, 4, 1, 0, 6, 9, 9, 6, 6, 8, 4, 8, 8, 6, 2, 4, 9, 8, 0, 3, 7, 6, 8, 7, 7, 1, 1

Offset: 0

Amiram Eldar, Jun 04 2021

			0.77750463411224827641758654542571050719247729622900...

Alvaro Meseguer, Fundamentals of Numerical Mathematics for Physicists and Engineers, Wiley, 2020, Chapter 4, exercise 12, p. 128.
John Michael Rassias, Geometry, Analysis, and Mechanics, World Scientific, 1994, p. 14.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 998.
Mathematics MI, Integral x/(e^x - 1) from 0 to 1, YouTube video, May 8 2021.
Voodooguru, Collaboration between Integration and Summation, Mathematical Meanderings, May 9 2021.
Eric Weisstein's World of Mathematics, Debye Functions.
Eric Weisstein's World of Mathematics, Dilogarithm.
Eric Weisstein's World of Mathematics, Polylogarithm.
Wikipedia, Debye function.
Wikipedia, Polylogarithm.
Wikipedia, Spence's function.

Cf. A027641, A027642, A120082, A120083.

evalf(-dilog(exp(1))-1/2, 140);  # Alois P. Heinz, Jun 04 2021

RealDigits[PolyLog[2, 1-1/E], 10, 100][[1]]

intnum(x=0, 1, x/(exp(x)-1)) \\ Michel Marcus, Jun 04 2021

Equals D_1(1) = Sum_{k>=0} A120082(k)/A120083(k), where D_n(x) are the Debye functions.
Equals Li_2(1-1/e) = -1/2 - Li_2(1-e) = Pi^2/6 - 1 + log(e-1) - Li_2(1/e), where Li_2(x) is the dilogarithm function.
Equals Sum_{k>=0} B(k)/(k+1)! = -1/2 + Sum_{k>=0} (-1)^k*B(k)/(k+1)! = -1/4 + Sum_{k>=0} B(2*k)/(2*k+1)!, where B(k) is the k-th Bernoulli number.
Equals Sum_{k>=1} (1 - (k+1)*exp(-k))/k^2.

cp's OEIS Frontend

A227830 Denominators of coefficients in expansion of x/(exp(x)-1).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Sage

Formula

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A182918 Denominators of the swinging Bernoulli number b_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A227540 Denominator of the rationals obtained from the e.g.f. D(1,x), a Debye function.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A341908 Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula