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%I A027641
%S A027641 1,1,1,0,1,0,1,0,1,0,5,0,691,0,7,0,3617,0,43867,0,174611,0,854513,
%T A027641 0,236364091,0,8553103,0,23749461029,0,8615841276005,0,7709321041217,
%U A027641 0,2577687858367,0,26315271553053477373,0,2929993913841559,0,261082718496449122051
%V A027641 1,-1,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,854513,
%W A027641 0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,-7709321041217,
%X A027641 0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N A027641 Numerator of Bernoulli number B_n.
%C A027641 B_{2n} = (-1)^(m-1)/2^(2m+1) * Integral{-inf..inf, [d^(m-1)/dx^(m-1) 
               sech(x)^2 ]^2 dx} (see Grosset/Veselov).
%C A027641 a(n)/A027642(n) (Bernoulli numbers) provide the a-sequence for the Sheffer 
               matrix A094816 (coefficients of orthogonal Poisson-Charlier polynomials). 
               See the W. Lang link under A006232 for a- and z-sequences for Sheffer 
               matrices. The corresponding z-sequence is given by the rationals 
               A130189(n)/A130190(n).
%C A027641 Harvey (2008) describes an algorithm for computing Bernoulli numbers. 
               Using a parallel implementation, he computes B(k) for k = 10^8, a 
               new record. His method is to compute B(k) modulo p for many small 
               primes p and then reconstruct B(k) via the Chinese Remainder Theorem. 
               The time complexity is O(k^2 log(k)^(2+epsilon)), matching that of 
               existing algorithms that exploit the relationship between B(k) and 
               zeta(k). An implementation of the new algorithm is significantly 
               faster than the implementations of the zeta-function method in PARI/
               GP and Mathematica. The algorithm is especially well-suited to parallelisation. 
               Some values, such as B(10^8) may be downloaded from his web site. 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 09 2008
%D A027641 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 A027641 H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. 
               Nach. 34 (1967), 377-378. Math Rev 36#4030.
%D A027641 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A027641 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 A027641 H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974; see 
               p. 11.
%D A027641 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.
%D A027641 Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, 
               Eulerian, MacMahon and Stirling number triangles, Journal of Integer 
               Sequences, Vol. 9 (2006), Article 06.4.1.
%D A027641 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; 
               Section 2.6.
%D A027641 L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
%D A027641 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 
               1.
%H A027641 T. D. Noe, <a href="b027641.txt">Table of n, a(n) for n=0..200</a>
%H A027641 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 A027641 M.-P. Grosset and A. P. Veselov, <a href="http://arXiv.org/abs/math.GM/
               0503175">Bernoulli numbers and solitons</a>
%H A027641 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer 
               Sequences, 4 (2001), #01.1.6.
%H A027641 K. Dilcher, <a href="http://www.mscs.dal.ca/%7Edilcher/bernoulli.html">
               A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise)</
               a>
%H A027641 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 A027641 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha134.htm">Factorizations of many number sequences</a>
%H A027641 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha1341.htm">Factorizations of many number sequences</a>
%H A027641 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 A027641 S. Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">
               The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]
%H A027641 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/
               euler/How%20Euler%20Did%20It%2023%20Bernoulli%20numbers.pdf">Bernoulli 
               numbers</a>
%H A027641 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BernoulliNumber.html">More information.</a>
%H A027641 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/
               BernoulliB/11">Generating functions of B_n & B_2n</a>
%H A027641 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%H A027641 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A027641 David Harvey, <a href="http://arxiv.org/pdf/0807.1347">A multimodular 
               algorithm for computing Bernoulli numbers</a>, July 8, 2008.
%H A027641 Peter Luschny, <a href="http://www.luschny.de/math/zeta/BernoulliEuler.pdf">
               Die Riemannsche Funktionalgleichung als Grundlage der Bernoulli und 
               Euler Funktion. </a>(2004) [From Peter Luschny (peter(AT)luschny.de), 
               May 02 2009]
%F A027641 E.g.f: x/(e^x - 1). Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j 
               as B_j).
%F A027641 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).
%F A027641 Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as 
               B_j).
%F A027641 B_{n-1} = - Sum_{r=1..n} (-1)^r binomial(n, r) r^(-1) Sum_{k=1..r} k^(n-1). 
               More concisely, B_n = 1 - (1-C)^(n+1), where C^r is replaced by the 
               arithmetic mean of the first r n-th powers of natural numbers in 
               the expansion of the right-hand side. [Bergmann]
%F A027641 Sum_{i=1..inf} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).
%F A027641 Let B(s,z) = -2^(1-s)(I/Pi)^s s! PolyLog(s,Exp(-2IPi/z)). Then B(2n,1) 
               = B_{2n} for n >= 1. Similarly the numbers B(2n+1,1) which might 
               be called Co-Bernoulli numbers can be considered and it is remarkable 
               that already Leonhard Euler in 1755 calculated B(3,1) and B(5,1) 
               (Opera Omnia, Ser. 1, Vol. 10, p. 351). (Cf. the Luschny reference 
               for a discussion.) [From Peter Luschny (peter(AT)luschny.de), May 
               02 2009]
%e A027641 B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 
               0, -691/2730, 0, 7/6, 0, -3617/510, ...
%p A027641 B := proc(n) sum( (-1)^'m'*'m'!*combinat[stirling2](n,'m')/('m'+1),'m'=0..n); 
               end;
%p A027641 B := proc(n) numtheory[bernoulli](n); end;
%p A027641 with(numtheory):seq(numer(bernoulli(n)) ,n=0..40);# [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 08 2009]
%t A027641 Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (from Robert G. Wilson 
               v Oct 11 2004)
%t A027641 Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 
               40}], x]]
%o A027641 (PARI) a(n)=if(n<0, 0, numerator(bernfrac(n)))
%Y A027641 This is the main entry for the Bernoulli numbers and has all the references, 
               links and formulae. Sequences A027642 (the denominators of B_n) and 
               A000367/A002445 = B_{2n} are also important!
%Y A027641 Cf. A027642, A000146, A000367, A002445.
%Y A027641 Cf. also A002882, A003245, A127187, A127188.
%Y A027641 Sequence in context: A157302 A036946 A164940 this_sequence A164555 A129205 
               A098173
%Y A027641 Adjacent sequences: A027638 A027639 A027640 this_sequence A027642 A027643 
               A027644
%K A027641 sign,frac,nice
%O A027641 0,11
%A A027641 N. J. A. Sloane (njas(AT)research.att.com).

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