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%I M4039 N1677 #187 Apr 03 2023 10:36:09
%S 1,1,-1,1,-1,5,-691,7,-3617,43867,-174611,854513,-236364091,8553103,
%T -23749461029,8615841276005,-7709321041217,2577687858367,
%U -26315271553053477373,2929993913841559,-261082718496449122051
%N Numerators of Bernoulli numbers B_2n.
%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 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D 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 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
%D F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 330.
%D H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Simon Plouffe, <a href="/A000367/b000367.txt">Table of n, a(n) for n = 0..249</a> [taken from link below]
%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 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.
%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 J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/miniature18.pdf">Some applications of Bernoulli numbers</a>
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/BernoulliNumber.html">Bernoulli number</a>
%H F. N. Castro, O. E. González, and L. A. Medina, <a href="http://emmy.uprrp.edu/lmedina/papers/eulerian/EulerianFinal.pdf">The p-adic valuation of Eulerian numbers: trees and Bernoulli numbers</a>, 2014.
%H R. Jovanovic, <a href="http://milan.milanovic.org/math/english/bernoulli/bernoulli.html">Bernoulli numbers and the Pascal triangle</a>
%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), #00.2.9.
%H B. C. Kellner, <a href="http://arXiv.org/abs/math.NT/0409223">On irregular prime power divisors of the Bernoulli numbers</a>, arXiv:math/0409223 [math.NT], 2004-2005.
%H B. C. Kellner, <a href="http://arxiv.org/abs/math/0411498">The structure of Bernoulli numbers</a>, arXiv:math/0411498 [math.NT], 2004.
%H C. Lin and L. Zhipeng, <a href="http://arXiv.org/abs/math.HO/0408082">On Bernoulli numbers and its properties</a>, arXiv:math/0408082 [math.HO], 2004.
%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) # 14.4.6.
%H H.-M. Liu, S-H. Qi, and S.-Y. Ding, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu/liu4.html">Some Recurrence Relations for Cauchy Numbers of the First Kind</a>, JIS 13 (2010) # 10.3.8.
%H S. O. S. Math, <a href="http://www.sosmath.com/tables/bernoulli/bernoulli.html">Bernoulli and Euler Numbers</a>
%H R. Mestrovic, <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), 14.8.4.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha134.htm">Factorizations of many number sequences</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1341.htm">Factorizations of many number sequences</a>
%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 N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
%H Ronald Orozco López, <a href="https://www.researchgate.net/publication/350397609_Solution_of_the_Differential_Equation_ykeay_Special_Values_of_Bell_Polynomials_and_ka-Autonomous_Coefficients">Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients</a>, Universidad de los Andes (Colombia 2021).
%H Simon Plouffe, <a href="http://plouffe.fr/simon/constants/ber250000.txt">The 250,000th Bernoulli Number</a>
%H Simon Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/Bernoulli/bernoulli1.html">Some Properties of Bernoulli's Numbers</a>
%H Zhi-Hong Sun, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.038">Congruences involving Bernoulli polynomials</a>, Discr. Math., 308 (2007), 71-112.
%H S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/bnum">Prime factors of the absolute values of Bernoulli numerators</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">Bernoulli Number.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</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 of even powers.
%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 If n >= 3 is prime, then 12*|a((n+1)/2)| == (-1)^((n-1)/2)*A002445((n+1)/2) (mod n). - _Vladimir Shevelev_, Sep 04 2010
%F a(n) = numerator(-i*(2*n)!/(Pi*(1-2*n))*Integral_{t=0..1} log(1-1/t)^(1-2*n) dt). - _Gerry Martens_, May 17 2011, corrected by _Vaclav Kotesovec_, Oct 22 2014
%F a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(1)) for n > 0. - _Peter Luschny_, Jun 29 2012
%F E.g.f.: G(0) where G(k) = 2*k + 1 - x*(2*k+1)/(x + (2*k+2)/(1 + x/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Feb 13 2013
%F a(n) = numerator(2*n*Sum_{k=0..2*n} (2*n+k-2)! * Sum_{j=1..k} ((-1)^(j+1) * Stirling1(2*n+j,j)) / ((k-j)!*(2*n+j)!)), n > 0. - _Vladimir Kruchinin_, Mar 15 2013
%F E.g.f.: E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 16 2013
%F E.g.f.: E(0) - x, where E(k) = x + k + 1 - x*(k+1)/E(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 14 2013
%F a(n) = numerator((-1)^(n+1)*2*Gamma(2*n + 1)*zeta(2*n)/(2*Pi)^(2*n)). - _Artur Jasinski_, Dec 29 2020
%F a(n) = numerator(-2*n*zeta(1 - 2*n)) for n > 0. - _Artur Jasinski_, Jan 01 2021
%e B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, ... ].
%p A000367 := n -> numer(bernoulli(2*n)):
%p # Illustrating an algorithmic approach:
%p S := proc(n,k) option remember; if k=0 then `if`(n=0,1,0) else S(n, k-1) + S(n-1, n-k) fi end: Bernoulli2n := n -> `if`(n = 0,1,(-1)^n * S(2*n-1,2*n-1)*n/(2^(2*n-1)*(1-4^n))); A000367 := n -> numer(Bernoulli2n(n)); seq(A000367(n),n=0..20); # _Peter Luschny_, Jul 08 2012
%t Numerator[ BernoulliB[ 2*Range[0, 20]]] (* _Jean-François Alcover_, Oct 16 2012 *)
%t Table[Numerator[(-1)^(n+1) 2 Gamma[2 n + 1] Zeta[2 n]/(2 Pi)^(2 n)], {n, 0, 20}] (* _Artur Jasinski_, Dec 29 2020 *)
%o (PARI) a(n)=numerator(bernfrac(2*n))
%o (Python) # The objective of this implementation is efficiency.
%o # n -> [a(0), a(1), ..., a(n)] for n > 0.
%o from fractions import Fraction
%o def A000367_list(n): # Bernoulli numerators
%o T = [0 for i in range(1, n+2)]
%o T[0] = 1; T[1] = 1
%o for k in range(2, n+1):
%o T[k] = (k-1)*T[k-1]
%o for k in range(2, n+1):
%o for j in range(k, n+1):
%o T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
%o a = 0; b = 6; s = 1
%o for k in range(1, n+1):
%o T[k] = s*Fraction(T[k]*k, b).numerator
%o h = b; b = 20*b - 64*a; a = h; s = -s
%o return T
%o print(A000367_list(100)) # _Peter Luschny_, Aug 09 2011
%o (Maxima)
%o B(n):=if n=0 then 1 else 2*n*sum((2*n+k-2)!*sum(((-1)^(j+1)*stirling1(2*n+j,j))/ ((k-j)!*(2*n+j)!),j,1,k),k,0,2*n);
%o makelist(num(B(n)),n,0,10); /* _Vladimir Kruchinin_, Mar 15 2013, fixed by _Vaclav Kotesovec_, Oct 22 2014 */
%Y B_n gives A027641/A027642. See A027641 for full list of references, links, formulas, etc.
%Y See A002445 for denominators.
%Y Cf. also A002882, A003245, A127187, A127188.
%K sign,frac,nice
%O 0,6
%A _N. J. A. Sloane_
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