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%I #133 Oct 27 2023 18:03:35
%S 1,-1,1,-1,1,-691,1,-3617,43867,-174611,77683,-236364091,657931,
%T -3392780147,1723168255201,-7709321041217,151628697551,
%U -26315271553053477373,154210205991661,-261082718496449122051,1520097643918070802691,-2530297234481911294093
%N Numerator of Bernoulli(2*n)/(2*n).
%C It was incorrectly claimed that a(n) is "also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(2*n*n!)"; actually, the numerators of these fractions and the numerators of "modified Bernoulli numbers" (see A057868 for details) differ from each other and from this sequence. - _Andrey Zabolotskiy_, Dec 03 2022
%C Ramanujan incorrectly conjectured that the sequence contains only primes (and 1). - _Jud McCranie_. See A112548, A119766.
%C a(n) = A046968(n) if n < 574; a(574) = 37 * A046968(574). - _Michael Somos_, Feb 01 2004
%C Absolute values give denominators of constant terms of Fourier series of meromorphic modular forms E_k/Delta, where E_k is the normalized k th Eisenstein series [cf. Gunning or Serre references] and Delta is the normalized unique weight-twelve cusp form for the full modular group (the generating function of Ramanujan's tau function.) - Barry Brent (barrybrent(AT)iphouse.com), Jun 01 2009
%C |a(n)| is a product of powers of irregular primes (A000928), with the exception of n = 1,2,3,4,5,7. - _Peter Luschny_, Jul 28 2009
%C Conjecture: If there is a prime p such that 2*n+1 < p and p divides a(n), then p^2 does not divide a(n). This conjecture is true for p < 12 million. - _Seiichi Manyama_, Jan 21 2017
%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 L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
%D R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
%D R. Kanigel, The Man Who Knew Infinity, pp. 91-92.
%D J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
%D J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, p. 93.
%H Seiichi Manyama, <a href="/A001067/b001067.txt">Table of n, a(n) for n = 1..314</a> (first 100 terms from T. D. Noe)
%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 D. Bar-Natan, T. T. Q. Le and D. P. Thurston, <a href="https://arxiv.org/abs/math/0204311">Two applications of elementary knot theory to Lie algebras and Vassiliev invariants</a>, arXiv:math/0204311 [math.QA], 2002-2003; Geometry and Topology 7-1 (2003) 1-31.
%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 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">MathWorld: Riemann Zeta Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">Bernoulli Number.</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture">Kummer-Vandiver conjecture</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.: numerators of coefficients of z^(2*n) in z/(exp(z)-1). - _Benoit Cloitre_, Jun 02 2003
%F For 2 <= k <= 1000 and k != 7, the 2-order of the full constant term of E_k/Delta = 3 + ord_2(k - 7). - Barry Brent (barrybrent(AT)iphouse.com), Jun 01 2009
%F G.f. for Bernoulli(2*n)/(2*n) = a(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n))*integral(log(1-1/t)^(2*n) dt,t=0,1). - _Gerry Martens_, May 18 2011
%F E.g.f.: a(n) = numerator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - _Peter Luschny_, Jul 12 2012
%F |a(n)| = numerator of Integral_{r=0..1} HurwitzZeta(1-n, r)^2 dr. More general: |Bernoulli(2*n)| = binomial(2*n,n)*n^2*I(n) for n >= 1 where I(n) denotes the integral. - _Peter Luschny_, May 24 2015
%e The sequence Bernoulli(2*n)/(2*n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ...
%e The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ...
%p A001067_list := proc(n) 1/(1-1/exp(z)); series(%,z,2*n+4);
%p seq(numer((2*i+1)!*coeff(%,z,2*i+1)),i=0..n) end:
%p A001067_list(21); # _Peter Luschny_, Jul 12 2012
%t Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (* _Robert G. Wilson v_, Feb 03 2004 *)
%o (PARI) {a(n) = if( n<1, 0, numerator( bernfrac(2*n) / (2*n)))}; /* _Michael Somos_, Feb 01 2004 */
%o (Sage)
%o @CachedFunction
%o def S(n, k) :
%o if k == 0 :
%o if n == 0 : return 1
%o else: return 0
%o return S(n, k-1) + S(n-1, n-k)
%o def BernoulliDivN(n) :
%o if n == 0 : return 1
%o return (-1)^n*S(2*n-1,2*n-1)/(4^n-16^n)
%o [BernoulliDivN(n).numerator() for n in (1..22)]
%o # _Peter Luschny_, Jul 08 2012
%o (Sage) [numerator(bernoulli(2*n)/(2*n)) for n in (1..25)] # _G. C. Greubel_, Sep 19 2019
%o (Magma) [Numerator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // _Vincenzo Librandi_, Sep 17 2015
%o (GAP) List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n))); # _G. C. Greubel_, Sep 19 2019
%Y Similar to but different from A046968. See A090495, A090496.
%Y Denominators given by A006953.
%Y Cf. A000367, A006863, A033563, A046968.
%Y Cf. A141590, A255505.
%K sign,frac,nice
%O 1,6
%A _N. J. A. Sloane_, Richard E. Borcherds (reb(AT)math.berkeley.edu)
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