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

0,13

Comments

Denominators are found under A120085.

This sequence appears to coincide with A120082.

Apparently a shifted version of A060054. - R. J. Mathar, May 22 2026

Links

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).

Formula

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

Example

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

Mathematica

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 *)

Prog

(Magma) [Numerator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023
(SageMath) [numerator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023

Crossrefs

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

Sequence in context: A214335 A060054 A120082 * A358625 A249699 A141588

Adjacent sequences: A120081 A120082 A120083 * A120085 A120086 A120087

Keyword

sign,frac

Author

Wolfdieter Lang, Jul 20 2006

Status

approved