%I M2039 #83 Sep 08 2022 08:44:35
%S 12,120,252,240,132,32760,12,8160,14364,6600,276,65520,12,3480,85932,
%T 16320,12,69090840,12,541200,75852,2760,564,2227680,132,6360,43092,
%U 6960,708,3407203800,12,32640,388332,120,9372,10087262640
%N a(n) = denominator of Bernoulli(2n)/(2n).
%C a(n) are alternately divisible by 12 and 120, a(n)/(12, 120, 12, 120, 12, 120, ...) = 1, 1, 21, 2, 11, 273, ... . - _Paul Curtz_, Sep 13 2011 and _Michel Marcus_, Jan 05 2013
%C A141590/(2 before a(n+1)) = 1/2 + 1/12 - 1/120 + 1/252 is an old semi-convergent series for Euler's constants A001620 ("2 before a" meaning that one term, namely 2, is inserted before the sequence). This series is discussed in details in reference [Blagouchine, 2016], Sect. 3 and Fig. 3. - _Paul Curtz_, Sep 13 2011, _Michel Marcus_, Jan 05 2013 and _Iaroslav V. Blagouchine_, Sep 16 2015
%C a(n) = A006863(n)/2. - _Michel Marcus_, Jan 05 2013
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 259, (6.3.18) and (6.3.19); also p. 810.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006953/b006953.txt">Table of n, a(n) for n=1..1000</a>
%H 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 [alternative scanned copy].
%H 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, p. 259, (6.3.18) and (6.3.19).
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2015.06.012">Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only</a>, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. <a href="http://arxiv.org/abs/1501.00740">arXiv version</a>, arXiv:1501.00740 [math.NT], 2015.
%H R. D. Carmichael, <a href="http://dx.doi.org/10.1090/S0002-9904-1909-01738-0">Notes on the simplex theory of numbers</a>, Bull. Amer. Math. Soc. 15 (1909), 217-223.
%H G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Ward/ward2.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
%H E. Z. Goren, <a href="http://www.math.mcgill.ca/goren/ZetaValues/Riemann.html">Tables of values of Riemann zeta functions</a>
%H A. Iványi, <a href="http://www.emis.de/journals/AUSM/C5-1/math51-5.pdf">Leader election in synchronous networks</a>, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">MathWorld: Riemann Zeta Function</a>
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">MathWorld: Harmonic Number</a>
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F Zeta(1-2*n) = -Bernoulli(2*n)/(2*n).
%F G.f. for Bernoulli(2*n)/(2*n) = A001067(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n)) * Integral_{t=0..1} log(1-1/t)^(2*n) dt. - _Gerry Martens_, May 18 2011
%F E.g.f.: a(n) = denominator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - _Peter Luschny_, Jul 12 2012
%e Sequence Bernoulli(2n)/(2n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ... .
%p A006953_list := proc(n) 1/(1-1/exp(z)); series(%,z,2*n+4);
%p seq(denom((-1)^i*(2*i+1)!*coeff(%,z,2*i+1)),i=0..n) end;
%p A006953_list(35); # _Peter Luschny_, Jul 12 2012
%t Table[Denominator[BernoulliB[2n]/(2n)],{n,40}] (* _Harvey P. Dale_, Jan 12 2022 *)
%o (Magma) [Denominator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // _Vincenzo Librandi_, Sep 17 2015
%o (PARI) a(n) = denominator(bernfrac(2*n)/(2*n)); \\ _Michel Marcus_, Apr 21 2016
%o (Sage) [denominator(bernoulli(2*n)/(2*n)) for n in (1..40)] # _G. C. Greubel_, Sep 19 2019
%o (GAP) List([1..40], n-> DenominatorRat(Bernoulli(2*n)/(2*n)) ); # _G. C. Greubel_, Sep 19 2019
%Y Numerators are given by A001067.
%K nonn,frac,easy,nice
%O 1,1
%A _Simon Plouffe_ and _N. J. A. Sloane_
%E Previous Mathematica program replaced by _Harvey P. Dale_, Jan 12 2022