%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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