0,11

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).

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).

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

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.

H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

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.

H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974; see p. 11.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.

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.

H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.

L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

T. D. Noe, Table of n, a(n) for n=0..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

K. Dilcher, A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise)

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.

S. Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]

E. Sandifer, How Euler Did It, Bernoulli numbers

Eric Weisstein's World of Mathematics, More information.

Wolfram Research, Generating functions of B_n & B_2n

Index entries for sequences related to Bernoulli numbers.

Index entries for "core" sequences

David Harvey, A multimodular algorithm for computing Bernoulli numbers, July 8, 2008.

Peter Luschny, Die Riemannsche Funktionalgleichung als Grundlage der Bernoulli und Euler Funktion. (2004) [From Peter Luschny (peter(AT)luschny.de), May 02 2009]

E.g.f: x/(e^x - 1). Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).

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).

Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).

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]

Sum_{i=1..inf} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).

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]

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, ...

B := proc(n) sum( (-1)^'m'*'m'!*combinat[stirling2](n, 'm')/('m'+1), 'm'=0..n); end;

B := proc(n) numtheory[bernoulli](n); end;

with(numtheory):seq(numer(bernoulli(n)) , n=0..40); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 08 2009]

Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (from Robert G. Wilson v Oct 11 2004)

Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]

(PARI) a(n)=if(n<0, 0, numerator(bernfrac(n)))

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!

Cf. A027642, A000146, A000367, A002445.

Cf. also A002882, A003245, A127187, A127188.

Sequence in context: A157302 A036946 A164940 this_sequence A164555 A129205 A098173

Adjacent sequences: A027638 A027639 A027640 this_sequence A027642 A027643 A027644

sign,frac,nice

N. J. A. Sloane (njas(AT)research.att.com).