A120080 -id:A120080 - cp's OEIS frontend

A120081 Denominators of expansion for original Debye function (n=3).

1, 8, 20, 1, 1680, 1, 90720, 1, 4435200, 1, 207567360, 1, 6538371840000, 1, 423437414400, 1, 67580611338240000, 1, 35763659520196608000, 1, 6155242080686899200000, 1, 117509166994931712000000, 1, 15244417230585693025075200000, 1, 1799300365026394374144000000

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are given in A120080.

See A120070 for the definition of the Debye function D(x)=D(3,x) and references and links.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000367, A002445, A120080.

[Denominator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023

max = 26; Denominator[CoefficientList[Integrate[Normal[Series[(3*(t^3/(Exp[t] -1)))/x^3, {t, 0, max}]], {t, 0, x}], x]] (* Jean-François Alcover, Oct 04 2011 *)
Table[Denominator[3*BernoulliB[n]/((n+3)*n!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

def A120081(n): return denominator(3*bernoulli(n)/((n+3)*factorial(n)))
[A120081(n) for n in range(51)] # G. C. Greubel, May 01 2023

a(n) = denominator(r(n)), with r(n) = [x^n]( 1 - 3*x/8 + Sum_{k >= 1} (3*B(2*k)/((2*k+3)*(2*k)!))*x^(2*k) ), where B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator( 3*Bernoulli(n)/((n+3)*n!) ), n >= 0. - G. C. Greubel, May 01 2023

A120086 Numerators of expansion of Debye function for n=4: D(4,x).

Original entry on oeis.org

1, -2, 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

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120087.

See the W. Lang link under A120080 for more details on the general case D(n,x), n= 1, 2, ... D(4,x) is the e.g.f. of the rational sequence {4*B(n)/(n+4)}, n >= 0. See A227573/A227574. - Wolfdieter Lang, Jul 17 2013

			Rationals r(n): [1, -2/5, 1/18, 0, -1/1440, 0, 1/75600, 0, -1/3628800, 0, 1/167650560, 0, -691/5230697472000, ...].

Vincenzo Librandi, Table of n, a(n) for n = 0..600
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^4)/4 extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A060054. [From R. J. Mathar, Aug 07 2008]
Cf. A000367/A002445, A027641/A027642, A120097, A227573/A227574 (D(4,x) as e.g.f.). - Wolfdieter Lang, Jul 17 2013
Cf. A120080, A120081, A120082, A120083, A120084, A120085, A120087 (denominators).

[Numerator(4*(n+1)*(n+2)*(n+3)*Bernoulli(n)/Factorial(n+4)): n in [0..50]]; // G. C. Greubel, May 02 2023

r[n_]:= 4*BernoulliB[n]/((n+4)*n!); Table[r[n]//Numerator, {n,0,36}] (* Jean-François Alcover, Jun 21 2013 *)

[numerator(4*(n+1)*(n+2)*(n+3)*bernoulli(n)/factorial(n+4)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = numerator(r(n)), with r(n) = [x^n](1 - 4*x/(2*(4+1)) + 2*Sum_{k >= 0} (B(2*k)/((k+2)*(2*k)!))*x^(2*k) ), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = numerator(4*B(n)/((n+4)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). From D(4,x) read as o.g.f. - Wolfdieter Lang, Jul 17 2013

A120087 Denominators of expansion of Debye function for n=4: D(4,x).

Original entry on oeis.org

1, 5, 18, 1, 1440, 1, 75600, 1, 3628800, 1, 167650560, 1, 5230697472000, 1, 336259123200, 1, 53353114214400000, 1, 28100018194440192000, 1, 4817145976189747200000, 1, 91657150256046735360000, 1, 11856768957122205686169600000, 1, 1396008903899788738560000000, 1

Offset: 0

Wolfdieter Lang, Jul 20 2006

From Wolfdieter Lang, Jul 17 2013: (Start)
The numerators are given in A120086.
See the link under A120080 for D(n,4) as e.g.f. of 4*B(n)/(n+4) = A227573(n)/A227574(n), n>= 0. (End)

			Rationals r(n): [1, -2/5, 1/18, 0, -1/1440, 0, 1/75600, 0, -1/3628800, 0, 1/167650560, 0, -691/5230697472000, ...].

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A000367, A002445, A027641, A027642, A120080, A120081, A120082, A120083, A120084, A120085, A120086, A227573, A227574.

[Denominator(4*(n+1)*(n+2)*(n+3)*Bernoulli(n)/Factorial(n+4)): n in [0..50]]; // G. C. Greubel, May 02 2023

Table[Denominator[4*BernoulliB[n]/((n+4)*n!)], {n,0,50}] (* G. C. Greubel, May 02 2023 *)

[denominator(4*(n+1)*(n+2)*(n+3)*bernoulli(n)/factorial(n+4)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = denominator(r(n)), with r(n) = [x^n](1 - 2*x/5 + 2*Sum_{k >= 0}(B(2*k)/((k+2)*(2*k)!))*x^(2*k) ), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator(4*B(n)/((n+4)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). From D(4,x) read as o.g.f. - Wolfdieter Lang, Jul 17 2013

A120084 Numerators of expansion for Debye function for n=2: D(2,x).

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

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120085.

This sequence appears to coincide with A120082.

			Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960, ...].

G. C. Greubel, 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^2)/2 extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A000367, A002445, A120080, A120081, A120082, A120083, A120085, A120086, A120087.

[Numerator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023

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 *)

[numerator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - x/3 + Sum_{k >= 1} (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

A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.

Original entry on oeis.org

1, -2, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 17 2013

The denominators are given in A227574.
For general remarks on the e.g.f.s D(n,x), the Debye function with index n = 1, 2, 3, ... see the W. Lang link under A120080.
D(4,x) := (4/x^4)*int(t^4/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(4,n) = 4*B(n)/(n+4), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).
See the Abramowitz-Stegun reference for the integral appearing in
  D(4,x) and a series expansion valid for |x| < 2*Pi.
Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Jul 19 2013
Differs from these sequences for n = 1328, 2660, 2828, 2880... - Andrey Zabolotskiy, Dec 08 2023

			The rationals r(4,n), n=0..15 are: 1, -2/5, 1/9, 0, -1/60, 0, 1/105, 0, -1/90, 0, 5/231, 0, -691/10920, 0, 7/27, 0.

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=4, with a factor (x^4)/4 extracted.

Cf. A227570, A227574, A027641/A027642, A120086/A120087 (D(4,x) as o.g.f.).

A227573[n_]:=Numerator[4BernoulliB[n]/(n+4)];
Array[A227573,50,0] (* Paolo Xausa, Dec 08 2023 *)

print([(bernoulli(n)*4/(n+4)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

a(n) = numerator(4*B(n)/(n+4)), n >= 0, with the Bernoulli numbers B(n).

A120085 Denominators of expansion for Debye function for n=2: D(2,x).

Original entry on oeis.org

1, 3, 24, 1, 2160, 1, 120960, 1, 6048000, 1, 287400960, 1, 9153720576000, 1, 597793996800, 1, 96035605585920000, 1, 51090942171709440000, 1, 8831434289681203200000, 1, 169213200472701665280000, 1, 22019713777512667702886400000, 1, 2605883287279605645312000000

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are found under A120084.
D(2,x) := (2/x^2)*Integral_{0..x} t^2/(exp(t)-1) dt is the e.g.f. of 2*B(n)/(n+2), n>=0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). Proof by using the e.g.f. for {k*B(k-1)} (with 0 for k=0) and integrating termwise (allowed for |x| <= r < rho with small enough rho).
See the Abramowitz-Stegun link for the integral and an expansion. - Wolfdieter Lang, Jul 16 2013

			Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960,...].

G. C. Greubel, Table of n, a(n) for n = 0..445
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=2, multiplied by 2/x^2.

Cf. A000367/A002445, A027641/A027642, A120080/A120081 (D(3,x) expansion), A120082/A120083 (D(1,x) expansion), A120084, A120086, A120087.

[Denominator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023

max = 25; Denominator[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[Denominator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* G. C. Greubel, May 02 2023 *)

[denominator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = denominator(r(n)), with r(n) = [x^n]( 1 - x/3 + Sum_{k >= 1} (B(2*k)/((k+1)*(2*k)!))*x^(2*k) ), |x|<2*pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) above. - Wolfdieter Lang, Jul 16 2013

A227570 Numerators of rationals with e.g.f. D(3,x), a Debye function.

Original entry on oeis.org

1, -3, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 16 2013

The denominators are given in A227571.
For general remarks on the e.g.f.s D(n,x), the Debye function with index n = 1, 2, 3, ... see the W. Lang link under A120080.
D(3,x) := (3/x^3)*int(t^3/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(3,n) = 3*B(n)/(n+3), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).
See the Abramowitz-Stegun link for the integral appearing in
  D(3,x) and a series expansion valid for |x| < 2*Pi.
Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Aug 13 2013
Differs from these sequences at n = 1292, 2624, 2770, 2778.... - Andrey Zabolotskiy, Dec 08 2023

			The rationals r(3,n), n=0..15 are: 1, -3/8, 1/10, 0, -1/70, 0, 1/126, 0, -1/110, 0, 5/286, 0, -691/13650, 0, 7/34, 0.

L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band V: Statistische Physik, Akademie Verlag, Leipzig, p. 195, equ. (63.5), and footnote 1 on p. 197.

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=3, with a factor (x^3)/3 extracted.

Cf. A227571, A227573, A027641/A027642, A120080/A120081 (D(3,x) as o.g.f.).

A227570[n_]:=Numerator[3BernoulliB[n]/(n+3)];
Array[A227570,50,0] (* Paolo Xausa, Dec 08 2023 *)

print([(bernoulli(n)*3/(n+3)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

a(n) = numerator(3*B(n)/(n+3)), n >= 0, with the Bernoulli numbers B(n).

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.

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A120081 Denominators of expansion for original Debye function (n=3).

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A120086 Numerators of expansion of Debye function for n=4: D(4,x).

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A120087 Denominators of expansion of Debye function for n=4: D(4,x).

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A120084 Numerators of expansion for Debye function for n=2: D(2,x).

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A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.

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A120085 Denominators of expansion for Debye function for n=2: D(2,x).

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A227570 Numerators of rationals with e.g.f. D(3,x), a Debye function.

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