A120083 -id:A120083 - cp's OEIS frontend

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

Wolfdieter Lang, Jul 20 2006

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

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

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

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).

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

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

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

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

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

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

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

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

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

A172282 Squares of Bernoulli number denominators A027642.

Original entry on oeis.org

1, 4, 36, 1, 900, 1, 1764, 1, 900, 1, 4356, 1, 7452900, 1, 36, 1, 260100, 1, 636804, 1, 108900, 1, 19044, 1, 7452900, 1, 36, 1, 756900, 1, 205119684, 1, 260100, 1, 36, 1, 3683290256100, 1, 36, 1, 183060900, 1, 3261636, 1, 476100, 1, 79524, 1, 2153888100, 1, 4356, 1, 2528100

Offset: 0

Paul Curtz, Jan 30 2010

Compare the sequence for example with A120083.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A120083, A172298.

Denominator[BernoulliB[Range[0,60]]]^2 (* Harvey P. Dale, Sep 26 2024 *)

A172282(n) = (denominator(bernfrac(n))^2); \\ Antti Karttunen, Dec 19 2018

a(n) = A027642(n)^2 .

Edited and extended by R. J. Mathar, Feb 02 2010

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.

A141588 Numerators of expansion for Debye function (D(1,x)) A120082 with 1's instead of 0's.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -691, 1, 1, 1, -3617, 1, 43867, 1, -174611, 1, 77683, 1, -236364091, 1, 657931, 1, -3392780147, 1, 1723168255201, 1, -7709321041217, 1, 151628697551, 1, -26315271553053477373

Offset: 0

Paul Curtz, Aug 20 2008

(Bernoulli numbers numerators) A027641(n)/a(n) is an integer sequence.

Note 1's in denominators A120083. A120082 and A027641 are analogous.

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; Numerator[ Table[a[n], {n, 0, 36}]] /. (0 -> 1) (* _Jean-François Alcover, Aug 09 2012 *)

Typo a(14)=7 instead of 1 fixed by Jean-François Alcover, Aug 09 2012

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

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

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

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A172282 Squares of Bernoulli number denominators A027642.

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