A227573 -id:A227573

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A120086

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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 [alternative scanned copy].

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).

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

sign,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

STATUS

approved

A120087

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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)

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

STATUS

approved

A227570

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..40.

FORMULA

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

EXAMPLE

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.

MATHEMATICA

A227570[n_]:=Numerator[3BernoulliB[n]/(n+3)];

Array[A227570, 50, 0] (* Paolo Xausa, Dec 08 2023 *)

PROG

(Sage)

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

CROSSREFS

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

KEYWORD

sign,easy,frac

AUTHOR

Wolfdieter Lang, Jul 16 2013

STATUS

approved

A227574

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

+10
4

1, 5, 9, 1, 60, 1, 105, 1, 90, 1, 231, 1, 10920, 1, 27, 1, 2550, 1, 4389, 1, 1980, 1, 897, 1, 19110, 1, 45, 1, 6960, 1, 121737, 1, 4590, 1, 57, 1, 19191900, 1, 63, 1, 148830, 1, 20769, 1, 8280, 1, 3525, 1, 603330, 1, 891, 1, 22260, 1, 11571, 1, 13050, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

See the comments and the Abramowitz-Stegun link under A227573.

LINKS

Table of n, a(n) for n=0..57.

FORMULA

a(n) = denominator(4*B(n)/(n+4)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).

The e.g.f. of the rationals r(4,n) := 4*B(n)/(n+4) is D(4,x) = (4/x^4)*int(t^4/(exp(t) - 1), t=0..x).

EXAMPLE

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.

MATHEMATICA

A227574[n_]:=Denominator[4BernoulliB[n]/(n+4)];

Array[A227574, 100, 0] (* Paolo Xausa, Dec 08 2023 *)

CROSSREFS

Cf. A227573, A027641/A027642, A120086/A120087 (D(4,x) as o.g.f.).

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 17 2013

STATUS

approved

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