This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120082 #97 May 01 2023 08:12:27
%S A120082 1,-1,1,0,-1,0,1,0,-1,0,1,0,-691,0,1,0,-3617,0,43867,0,-174611,0,
%T A120082 77683,0,-236364091,0,657931,0,-3392780147,0,1723168255201,0,
%U A120082 -7709321041217,0,151628697551,0,-26315271553053477373,0,154210205991661,0,-261082718496449122051
%N A120082 Numerators of expansion for Debye function for n=1: D(1,x).
%C A120082 Denominators are found under A120083.
%C A120082 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)}_{n>=0}. See A027641(n)/A227540(n). Thanks to _Peter Luschny_ for asking me to revisit this sequence. - _Wolfdieter Lang_, Jul 15 2013
%C A120082 Also numerators of coefficients in expansion of x/(exp(x)-1). See A227830 for denominators. - _N. J. A. Sloane_, Aug 01 2013
%D A120082 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 23.
%H A120082 Peter Luschny, <a href="/A120082/b120082.txt">Table of n, a(n) for n = 0..500</a>
%H A120082 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, 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.
%H A120082 Wolfdieter Lang, <a href="/A120082/a120082.pdf">Rationals r(n)</a>.
%F A120082 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).
%F A120082 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
%F A120082 Apart from the sign of a(1) this sequence differs from A358625 for the first time at n = 68. - _Peter Luschny_, Dec 02 2022
%e A120082 Rationals r(n): [1, -1/4, 1/36, 0, -1/3600, 0, 1/211680, 0, -1/10886400, ...].
%p A120082 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
%p A120082 gf := (1 - x/4 + sum((bernoulli(2*k)/((2*k+1)*(2*k)!))*x^(2*k), k=0..infinity)):
%p A120082 a := proc(n) local ser; if n = 0 then return 1 fi; ser := series(gf, x, n+2):
%p A120082 numer(coeff(ser, x, n)) end: seq(a(n), n = 0..40); # _Peter Luschny_, Dec 02 2022
%t A120082 Table[Numerator[BernoulliB[n]/((n+1)!)], {n,0,50}] (* _G. C. Greubel_, May 01 2023 *)
%o A120082 (Magma) [Numerator(Bernoulli(n)/Factorial(n+1)): n in [0..50]]; // _G. C. Greubel_, May 01 2023
%o A120082 (SageMath)
%o A120082 def A120082(n): return numerator(bernoulli(n)/factorial(n+1))
%o A120082 [A120082(n) for n in range(51)] # _G. C. Greubel_, May 01 2023
%Y A120082 Cf. A060054, A060055, A120083, A182918, A227540, A227830, A358625.
%K A120082 sign,frac
%O A120082 0,13
%A A120082 _Wolfdieter Lang_, Jul 20 2006
%E A120082 Edited after _Andrey Zabolotskiy_ noticed an inconsistency by _Peter Luschny_, Dec 02 2022