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%I #257 Mar 22 2024 08:52:09
%S 1,-1,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,
%T 854513,0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,
%U -7709321041217,0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N Numerator of Bernoulli number B_n.
%C 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 Harvey (2008) describes a new algorithm for computing Bernoulli numbers. 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+eps)). The algorithm is especially well-suited to parallelisation. - _Jonathan Vos Post_, Jul 09 2008
%C Regard the Bernoulli numbers as forming a vector = B_n, and the variant starting (1, 1/2, 1/6, 0, -1/30, ...), (i.e., the first 1/2 has sign +) as forming a vector Bv_n. The relationship between the Pascal triangle matrix, B_n, and Bv_n is as follows: The binomial transform of B_n = Bv_n. B_n is unchanged when multiplied by the Pascal matrix with rows signed (+-+-, ...), i.e., (1; -1,-1; 1,2,1; ...). Bv_n is unchanged 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
%C The sequence of the Bernoulli numbers B_n = a(n)/A027642(n) is the inverse binomial transform of the sequence {A164555(n)/A027642(n)}, illustrated by the fact that they appear as top row and left column in A190339. - _Paul Curtz_, May 13 2016
%C Named by de Moivre (1773; "the numbers of Mr. James Bernoulli") after the Swiss mathematician Jacob Bernoulli (1655-1705). - _Amiram Eldar_, Oct 02 2023
%D 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 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D Harold 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 Harold M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974; see p. 11.
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.
%D Herman H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
%D L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
%D Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%H T. D. Noe, <a href="/A027641/b027641.txt">Table of n, a(n) for n = 0..200</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H C. M. Bender and K. A. Milton, <a href="http://arxiv.org/abs/hep-th/9304052">Continued fraction as a discrete nonlinear transform</a>, arXiv:hep-th/9304052, 1993.
%H Beáta Bényi and Péter Hajnal, <a href="https://arxiv.org/abs/1804.01868">Poly-Bernoulli Numbers and Eulerian Numbers</a>, arXiv:1804.01868 [math.CO], 2018.
%H H. Bergmann, <a href="http://dx.doi.org/10.1002/mana.1967.3210340509">Eine explizite Darstellung der Bernoullischen Zahlen</a>, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.
%H Richard P. Brent and David Harvey, <a href="http://arxiv.org/abs/1108.0286">Fast computation of Bernoulli, Tangent and Secant numbers</a>, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
%H K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/CHEN/AlgBE2.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H W. Y.C. Chen, J. J. F. Guo and L. X. W. Wang, <a href="http://arxiv.org/abs/1208.5213">Log-behavior of the Bernoulli Numbers</a>, arXiv:1208.5213 [math.CO], 2012-2013.
%H Abraham de Moivre, <a href="https://archive.org/details/doctrineofchance00moiv/page/94/mode/2up">The Doctrine of Chances</a>, 3rd edition, London, 1733, p. 95.
%H K. Dilcher, <a href="http://www.mscs.dal.ca/%7Edilcher/bernoulli.html">A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise)</a>
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H H. W. Gould and J. Quaintance, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Quaintance/quain3.html">Bernoulli Numbers and a New Binomial Transform Identity</a>, J. Int. Seq. 17 (2014), Article 14.2.2.
%H M.-P. Grosset and A. P. Veselov, <a href="http://arXiv.org/abs/math.GM/0503175">Bernoulli numbers and solitons</a>, arXiv:math/0503175 [math.GM], 2005.
%H David Harvey, <a href="http://arxiv.org/abs/0807.1347">A multimodular algorithm for computing Bernoulli numbers</a>, arXiv:0807.1347 [math.NT], Jul 08 2008.
%H A. Iványi, <a href="http://www.emis.de/journals/AUSM/C5-1/math51-5.pdf">Leader election in synchronous networks</a>, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
%H M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), Article 00.2.9.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Srivastava/sriva3.html">Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers</a>, J. Int. Seq. 17 (2014), Article 14.4.6.
%H F. Luca and P. Stanica, <a href="http://calhoun.nps.edu/bitstream/handle/10945/29605/LucaStanicaJCNTfinal.pdf?sequence=1">On some conjectures on the monotonicity of some arithmetical sequences</a>, J. Combin. Number Theory 4 (2012) 1-10.
%H Peter Luschny, <a href="http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html">The Bernoulli Manifesto. A survey on the occasion of the 300th anniversary of the publication of Jacob Bernoulli's Ars Conjectandi, 1713-2013</a>.
%H Romeo Meštrović, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha134.htm">Bernoulli numbers (n = 2 to 114)</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1341.htm">(n = 116 to 154)</a> (Factorizations).
%H A. F. Neto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Neto/neto7.html">Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra</a>, J. Int. Seq. 18 (2015), Article 15.5.6.
%H A. F. Neto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Neto/neto10.html">A note of a Theorem of Guo, Mezo, and Qi</a>, J. Int. Seq. 19 (2016) Article 16.4.8.
%H Niels Nielsen, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k62119c">Traite Elementaire des Nombres de Bernoulli</a>, Gauthier-Villars, 1923, pp. 398.
%H Simon Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a>. [Project Gutenberg Etext]
%H Ed Sandifer, How Euler Did It, <a href="https://www.maa.org/sites/default/files/pdf/editorial/euler/How%20Euler%20Did%20It%2023%20Bernoulli%20numbers.pdf">Bernoulli numbers</a>.
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5.
%H J. Singh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Singh/singh8.html">On an Arithmetic Convolution</a>, J. Int. Seq. 17 (2014) Article 14.6.7.
%H J. Sondow and E. Tsukerman, <a href="https://arxiv.org/abs/1401.0322">The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers</a>, arXiv:1401.0322 [math.NT], 2014; see section 5.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">Bernoulli Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%H Roman Witula, Damian Slota and Edyta Hetmaniok, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from255to263.pdf">Bridges between different known integer sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.
%H Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/BernoulliB/11">Generating functions of B_n & B_2n</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%F E.g.f: x/(exp(x) - 1); take numerators.
%F Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).
%F B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k>=1} 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 Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).
%F 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 Sum_{i>=1} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).
%F 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).
%F Let B(s,z) = -2^(1-s)(i/Pi)^s s! PolyLog(s,exp(-2*i*Pi/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 Leonhard Euler in 1755 already calculated B(3,1) and B(5,1) (Opera Omnia, Ser. 1, Vol. 10, p. 351). (Cf. the Luschny reference for a discussion.) - _Peter Luschny_, May 02 2009
%F 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
%F From _Sergei N. Gladkovskii_, Dec 04 2012: (Start)
%F 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.
%F 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)
%F 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
%F 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
%F 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
%F 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
%F 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) log(Gamma(z)). - _Vladimir Reshetnikov_, Apr 24 2013
%F 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
%F B_n = Sum_{m=0..n} (-1)^m *A131689(n, m)/(m + 1), n >= 0. See one of the Maple programs. - _Wolfdieter Lang_, May 05 2017
%F a(n) = numerator((-1)^n*A155585(n-1)*n/(4^n-2^n)), for n>=1. - _Mats Granvik_, Nov 26 2017
%F From _Artur Jasinski_, Dec 30 2020: (Start)
%F a(n) = numerator(-2*cos(Pi*n/2)*Gamma(n+1)*zeta(n)/(2*Pi)^n), for n=0 and n>1.
%F a(n) = numerator(-n*zeta(1-n)), for n=0 and n>1. (End)
%e 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 B := n -> add((-1)^m*m!*Stirling2(n, m)/(m+1), m=0..n);
%p B := n -> bernoulli(n);
%p seq(numer(bernoulli(n)), n=0..40); # _Zerinvary Lajos_, Apr 08 2009
%t Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (* _Robert G. Wilson v_, Oct 11 2004 *)
%t Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]
%t Numerator[CoefficientList[Series[PolyGamma[1, 1/x]/x - x, {x, 0, 40}, Assumptions -> x > 0], x]] (* _Vladimir Reshetnikov_, Apr 24 2013 *)
%o (PARI) a(n)=numerator(bernfrac(n))
%o (Maxima) B(n):=(-1)^((n))*sum((stirling1(n,k)*stirling2(n+k,n))/binomial(n+k,k),k,0,n);
%o makelist(num(B(n)),n,0,20); /* _Vladimir Kruchinin_, Mar 16 2013 */
%o (Magma) [Numerator(Bernoulli(n)): n in [0..40]]; // _Vincenzo Librandi_, Mar 17 2014
%o (Sage)
%o [bernoulli(n).numerator() for n in range(41)] # _Peter Luschny_, Feb 19 2016
%o (Sage) # Alternatively:
%o def A027641_list(len):
%o f, R, C = 1, [1], [1]+[0]*(len-1)
%o for n in (1..len-1):
%o f *= n
%o for k in range(n, 0, -1):
%o C[k] = C[k-1] / (k+1)
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append((C[0]*f).numerator())
%o return R
%o A027641_list(41) # _Peter Luschny_, Feb 20 2016
%o (Python)
%o from sympy import bernoulli
%o from fractions import Fraction
%o [bernoulli(i).as_numer_denom()[0] for i in range(51)] # _Indranil Ghosh_, Mar 18 2017
%o (Python)
%o from sympy import bernoulli
%o def A027641(n): return bernoulli(n).p
%o print([A027641(n) for n in range(80)]) # _M. F. Hasler_, Jun 11 2019
%Y This is the main entry for the Bernoulli numbers and has all the references, links and formulas. Sequences A027642 (the denominators of B_n) and A000367/A002445 = B_{2n} are also important!
%Y A refinement is A194587.
%Y Cf. A000146, A002882, A003245, A127187, A127188, A074909, A164555, A176327, A190339, A131689.
%K sign,frac,nice,core
%O 0,11
%A _N. J. A. Sloane_
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