A227574 -id:A227574 - cp's OEIS frontend

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

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

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).

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

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Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Sage

Formula