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%I A000367 M4039 N1677
%S A000367 1,1,1,1,1,5,691,7,3617,43867,174611,854513,236364091,8553103,
%T A000367 23749461029,8615841276005,7709321041217,2577687858367,
%U A000367 26315271553053477373,2929993913841559,261082718496449122051
%V A000367 1,1,-1,1,-1,5,-691,7,-3617,43867,-174611,854513,-236364091,8553103,
%W A000367 -23749461029,8615841276005,-7709321041217,2577687858367,
%X A000367 -26315271553053477373,2929993913841559,-261082718496449122051
%N A000367 Numerators of Bernoulli numbers B_2n.
%D A000367 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 810.
%D A000367 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A000367 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd 
               ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity 
               Univ., San Antonio, TX, Vol. 2, p. 230.
%D A000367 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000367 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; 
               Section 2.6.
%D A000367 F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 
               2000, p. 330.
%D A000367 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 
               1.
%D A000367 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000367 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000367 Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 
               308 (2007), 71-112.
%H A000367 S. Plouffe, <a href="b000367.txt">Table of n, a(n) for n = 0..249</a> 
               [taken from link below]
%H A000367 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000367 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent 
               and Bernoulli numbers</a> related to Motzkin and Catalan numbers 
               by means of numerical triangles.
%H A000367 J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/
               miniature18.pdf">Some applications of Bernoulli numbers</a>
%H A000367 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php/BernoulliNumber.html">Bernoulli number</a>
%H A000367 R. Jovanovic, <a href="http://milan.milanovic.org/math/english/bernoulli/
               bernoulli.html">Bernoulli numbers and the Pascal triangle</a>
%H A000367 M. Kaneko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer 
               Sequences, 3 (2000), #00.2.9.
%H A000367 B. C. Kellner, <a href="http://arXiv.org/abs/math.NT/0409223">On irregular 
               prime power divisors of the Bernoulli numbers</a>
%H A000367 B. C. Kellner, <a href="http://arXiv.org/math.NT/0411498">The structure 
               of Bernoulli numbers</a>
%H A000367 C. Lin and L. Zhipeng, <a href="http://arXiv.org/abs/math.HO/0408082">
               On Bernoulli numbers and its properties</a>
%H A000367 S. O. S. Math, <a href="http://www.sosmath.com/tables/bernoulli/bernoulli.html">
               Bernoulli and Euler Numbers</a>
%H A000367 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha134.htm">Factorizations of many number sequences</a>
%H A000367 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha1341.htm">Factorizations of many number sequences</a>
%H A000367 Niels Nielsen, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=N062119">
               Traite Elementaire des Nombres de Bernoulli</a>, Gauthier-Villars, 
               1923, pp. 398.
%H A000367 S. Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/ber250000.txt">
               The 250,000-th Bernoulli Number</a>
%H A000367 S. Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">
               The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]
%H A000367 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/
               Bernoulli/bernoulli1.html">Some Properties of Bernoulli's Numbers</
               a>
%H A000367 S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/
               bnum">Prime factors of the absolute values of Bernoulli numerators</
               a>
%H A000367 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BernoulliNumber.html">More information.</a>
%H A000367 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli 
               number</a>
%H A000367 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%F A000367 E.g.f: t/(e^t - 1).
%F A000367 B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives 
               asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} 
               ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).
%e A000367 B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510,
               ... ].
%p A000367 bernoulli(n);
%o A000367 (PARI) a(n)=numerator(bernfrac(2*n))
%Y A000367 B_n gives A027641/A027642. See A027641 for full list of references, links, 
               formulae, etc.
%Y A000367 See A002445 for denominators.
%Y A000367 Cf. also A002882, A003245, A127187, A127188.
%Y A000367 Sequence in context: A117709 A133750 A090947 this_sequence A092133 A071772 
               A157281
%Y A000367 Adjacent sequences: A000364 A000365 A000366 this_sequence A000368 A000369 
               A000370
%K A000367 sign,frac,nice
%O A000367 0,6
%A A000367 N. J. A. Sloane (njas(AT)research.att.com).

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