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A027641 Numerator of Bernoulli number B_n. 121
1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

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, Jul 09 2008

Let the Bernoulli numbers as a vector = B_n, and the variant starting (1, 1/2, 1/6, 0, -1/30,...), (i.e. the first 1/2 is signed (+)) = Bv_n.  The relationship between the Pascal's triangle matrix, B_n, and Bv_n is as follows: The binomial transform of B_n = Bv_n. B_n is invariant when multiplied by the Pascal matrix with rows signed (+-+-,...), i.e. (1; -1,-1; 1,2,1;...). Bv_n is invariant when multiplied by the Pascal matrix with columns signed (+-+-,...), i.e. (1; 1,-1; 1,-2,1; 1,-3,3,-1;...). - Gary W. Adamson, Jun 29 2012

REFERENCES

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.

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

F. Luca and P. Stanica, On some conjectures on the monotonicity of some arithmetical sequences, J. Combin. Number Theory 4 (2012) 1-10.

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

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

LINKS

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].

C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304052, 1993.

Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, Arxiv preprint arXiv:1108.0286, 2011

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

W. Y.C. Chen, J. J. F. Guo and L. X. W. Wang, Log-behavior of the Bernoulli Numbers, arXiv:1208.5213.

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

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.

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

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

A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.

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

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

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.

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

E. Sandifer, How Euler Did It, Bernoulli numbers

J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.

Eric Weisstein's World of Mathematics, Bernoulli Number.

Eric Weisstein's World of Mathematics, Polygamma Function

Roman Witula, Damian Slota and Edyta Hetmaniok, Bridges between different known integer sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.

Wolfram Research, Generating functions of B_n & B_2n

Index entries for sequences related to Bernoulli numbers.

Index entries for "core" sequences

FORMULA

E.g.f: x/(exp(x) - 1); take numerators.

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)!).

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).

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, May 02 2009]

The B_n sequence is the left column of the inverse of triangle A074909, the "beheaded" Pascal's triangle. - Gary W. Adamson, Mar 05 2012

From  Sergei N. Gladkovskii, Dec 04 2012. (Start)

E.g.f. E(x)= 2 - x/(tan(x)+sec(x)-1)= sum (n>=0, a(n)*x^n/n!), a(n)=|B(n)|, where B(n) is Bernoulli number B_n.

E(x)= 2 + x - B(0), where B(k)=  4*k+1 + x/(2 + x/(4*k+3 - x/(2 - x/B(k+1) )));(continued fraction, 4-step).(End)

E.g.f.: x/(exp(x)-1)= U(0); U(k)= 2*k+1 - x(2*k+1)/(x + (2*k+2)/(1 + x/U(k+1) ));(continued fraction). - Sergei N. Gladkovskii, Dec 05 2012

E.g.f.: 2*(x-1)/(x*Q(0)-2) where Q(k) =  1 + 2*x*(k+1)/((2*k+1)*(2*k+3) - x*(2*k+1)*(2*k+3)^2/(x*(2*k+3) + 4*(k+1)*(k+2)/Q(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 26 2013

a(n)=numerator(B(n)), B(n)=(-1)^((n))*sum(k=0..n, (stirling1(n,k) * stirling2(n+k,n)) / binomial(n+k,k)). [Vladimir Kruchinin, Mar 16 2013]

E.g.f.: x/(exp(x)-1) = E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013

G.f. for Bernoulli(n) = a(n)/A027642(n): psi_1(1/x)/x - x, where psi_n(z) is the polygamma function, psi_n(z) = (d/dz)^{n+1} ln(Gamma(z)). [Vladimir Reshetnikov, Apr 24 2013]

E.g.f.: 2*E(0) - 2*x, where E(k)= x + (k+1)/(1 + 1/(1 - x/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013

EXAMPLE

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, ...

MAPLE

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, Apr 08 2009]

MATHEMATICA

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

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

Numerator[CoefficientList[Series[PolyGamma[1, 1/x]/x - x, {x, 0, 40}, Assumptions -> x > 0], x]] (* Vladimir Reshetnikov, Apr 24 2013 *)

PROG

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

(Maxima)

B(n):=(-1)^((n))*sum((stirling1(n, k)*stirling2(n+k, n))/binomial(n+k, k), k, 0, n);

makelist(num(B(n), n, 0, 20); [Vladimir Kruchinin, Mar 16 2013]

(MAGMA) [Numerator(Bernoulli(n)): n in [0..40]]; // Vincenzo Librandi, Mar 17 2014

CROSSREFS

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, A002882, A003245, A127187, A127188.

Cf. A074909.

Sequence in context: A036946 * A164555 A176327 A226156 A215616 A129205

Adjacent sequences:  A027638 A027639 A027640 * A027642 A027643 A027644

KEYWORD

sign,frac,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 27 15:22 EDT 2014. Contains 246143 sequences.